UC-NRLF 


REESE  LIBRARY 


UNIVERSITY  OF  CALIFORNIA. 

Co 


^Accession  No .       o  o  0  5  9  .   Class  No . 


SCIENTIFIC    MEMOIRS 

EDITED  BY 

J.  S.  AMES,  PH.D. 

PROFESSOR    OF    PHYSICS    IN    JOHNS    HOPKINS    UNIVERSITY 


X. 
THE  WAVE-THEORY  OF  LIGHT 


THE 

WAVE  THEORY  OF   LIGHT 

MEMOIRS  BY  HUYGENS,  YOUNG 
AND  FRESNEL 


EDITED  BY 

HENRY    CREW,   Pn.D. 

\N 

PROFKSSOR   OK   PHYSICS,  NORTHWESTERN    UNIVERSITY 


NEW  YORK    •:•    CINCINNATI    .:•    CHICAGO 

AMERICAN    BOOK    COMPANY 


to  *i<& 


COPYRIGHT,  1900,  HY 
AMERICAN   BOOK  COMPANY 


Crew,  Light. 
W.  P.   I 


PKEFACE 


THANKS  to  the  labors  of  Kirchhoff,  Kelvin,  Huxley,  and 
others,  there  is  now  a  widespread  opinion  that  any  physical 
phenomenon  is  "  explained"  only  when  some  one  has  devised  a 
dynamical  model  which  will  duplicate  the  phenomenon.  The 
completeness  of  the  explanation  is  to  be  measured  by  the  com- 
pleteness with  which  the  model  will  duplicate  the  phenomenon. 
Thus,  for  instance,  a  refraction  model  which,  like  that  of 
Airy,  describes  only  the  path  of  the  refracted  ray  when  the 
incident  ray  is  given,  does  not  in  any  true  sense  explain  how 
the  refracted  ray  comes  to  take  one  path  rather  than  another. 
Such  a  model  illustrates  Suell's  law,  but  does  not  explain  the 
phenomenon. 

If,  however,  we  take  a  large  and  shallow  tank  of  water,  the 
floor  of  the  tank  being  partly  covered  with  a  false  bottom,  so 
as  to  give  two,  and  only  two,  different  depths  of  water,  we  shall 
find  that  the  speed  of  the  waves  in  the  deeper  portion  of  the 
tank  bears  to  the  speed  in  the  shallower  portion  a  constant 
ratio  ;  hence,  in  passing  from  one  depth  to  the  other,  these 
waves  are  refracted  according  to  the  sine  law. 

Such  a  model  may  be  said  to  be  a  "  partial  explanation"  of 
refraction  in  so  far  as  it  refers  the  phenomenon  to  change  of 
speed  which  accompanies  change  of  medium.  It  represents, 
however,  only  the  kinematics  of  refraction. 

If,  now,  we  could  go  one  step  further,  and  make  a  model  in 
which  the  wave -producing  forces  were  duplicated — in  other 
words,  if  we  could  make  a  model  of  the  medium  and  of  the 
disturbing  forces — we  should  have  a  fairly  complete  "expla- 
nation" of  refraction ;  in  fact,  the  dynamics  of  refraction  would 
be  understood.  This  would  imply  not  only  that  we  knew  the 
substance  disturbed,  but  also  that  we  were  acquainted  with 
the  laws  according  to  which  it  is  disturbed. 


83059 


PREFACE 

A  theory  of  light  may  be  considered  either  from  a  kinemat- 
ical  or  from  a  dynamical  point  of  view.  To  assume,  on  exper- 
imental grounds,  that  a  ray  of  light  has  a  certain  speed  in  one 
medium  and  a  different  speed  in  a  different  medium,  and  that 
it  consists  in  a  particular  kind  of  motion,  and  thence  to  infer 
the  laws  of  refraction,  rectilinear  propagation,  and  diffraction, 
is  to  construct  a  kinematical  theory  of  light.  But  to  assume 
a  certain  structure  for  the  luminous  body  and  for  the  medium, 
and  thence  to  derive  the  motions  and  the  different  speeds 
assumed  in  the  kinematical  case,  is  to  offer  a  dynamical  ex- 
planation of  light. 

The  wave-theory  of  light  is  used,  nearly  always,  in  the  former 
and  narrower  sense  to  mean  the  kinematical  explanation  of 
light;  it  leaves  entirely  to  one  side  the  dynamical  questions 
hinted  at  above.  It  assumes,  not  without  strong  experimental 
evidence,  the  existence  of  waves  travelling  with  different  speeds 
in  different  media,  and  proposes  to  explain  the  cardinal  phe- 
nomena of  optics. 

To  illustrate  its  limitations,  we  may  cite  the  instance  of  the 
ordinary  and  extraordinary  ray  in  crystals.  How  it  happens 
that  there  are  two  rays  is  a  problem  in  the  dynamics  of  light ; 
but,  assuming  these  two  rays,  their  subsequent  behavior,  their 
inability  to  interfere,  etc.,  must  be  accounted  for — in  a  general 
way,  at  least — by  the  kinematical  theory  of  light. 

It  is  in  this  narrow  sense  that  the  wave-theory  of  light  is 
employed  in  the  memoirs  translated  in  this  volume. 

The  first  clear  and  unmistakable  suggestion  that  light  con- 
sists in  a  vibratory  motion  appears  to  be  due  to  that  brilliant 
but  unfortunate  genius,  Robert  Hooke  (1635-1703),  who,  in 
his  Micrograpliia  (London,  1665),  describes  the  three  charac- 
teristic features  of  the  motion  which  he  believes  to  constitute 
light. 

Since  it  has  not  been  deemed  advisable  to  reprint  Hooke's 
paper  in  this  volume,  it  may  not  be  out  of  place  here  to  quote 
what  few  paragraphs  are  necessary  fairly  to  present  his  point 
of  view.  This  will,  perhaps,  be  accomplished  by  the  follow- 
ing selections  : 

' '  It  would  be  somewhat  too  long  a  work  for  this  place  Zetet- 
ically  to  examine,  and  positively  to  prove,  what  particular 
kind  of  motion  it  is  that  must  be  the  efficient  of  Light;  for 
though  it  be  a  motion,  yet  'tis  not  every  motion  that  produces 

vi 


PREFACE 

it,  since  we  find  there  are  many  bodies  very  violently  mov'd, 
which  yet  afford  not  such  an  effect ;  and  there  are  other  bodies, 
which  to  our  senses,  seem  not  mov'd  so  much,  which  yet  shine. 
Thus  Water  and  quick-silver,  and  most  other  liquors  heated, 
shine  not;  and  several  hard  bodies,  as  Iron,  Silver,  Brass,  Cop- 
per, Wood,  &c.,  though  very  often  struck  with  a  hammer,  shine 
not  presently,  though  they  will  all  of  them  grow  exceeding 
hot;  whereas  rotten  Wood,  rotten  Fish,  Sea  Water,  Gloworms, 
&G.  have  nothing  of  tangible  heat  in  them,  and  yet  (where 
there  is  no  stronger  light  to  affect  the  Sensory)  they  shine  some 
of  them  so  Vividly,  that  one  may  make  a  shift  to  read  by  them. 

"It  would  be  too  long,  I  say,  here  to  insert  the  discursive 
progress  by  which  I  inquir'd  after  the  proprieties  of  the  mo- 
tion of  Light,  and  therefore  I  shall  only  add  the  result. 

"And,  First,  I  found  it  ought  to  be  exceeding  quick,  such 
as  those  motions  of  fermentation  and  putrefaction,  whereby, 
certainly,  the  parts  are  exceeding  nimbly  and  violently  mov'd; 
and  that,  because  we  find  those  motions  are  able  more  mi- 
nutely to  shatter  and  divide  the  body,  then  the  most  violent 
heats  or  menstruums  we  yet  know.  And  that  fire  is  nothing 
else  but  such  a  dissolution  of  the  Burning  body,  made  by  the 
most  universal  menstruum  of  all  sulphureous  bodies,  namely, 
the  Air,  we  shall  in  an  other  place  of  this  Tractate  endeavour 
to  make  probable.  And  that,  in  all  extremely  hot  shining 
bodies,  there  is  a  very  quick  motion  that  causes  Light,  as  well 
as  a  more  robust  that  causes  Heat,  may  be  argued  from  the 
celerity  wherewith  the  bodyes  are  dissolv'd. 

"Next,  it  must  be  a  Vibrative  motion.  And  for  this  the 
newly  mentioned  Diamond  affords  us  a  good  argument;  since 
if  the  motion  of  the  parts  did  not  return,  the  Diamond  must 
after  many  rubbings  decay  and  be  wasted  ;  but  we  have  no 
reason  to  suspect  the  latter,  especially  if  we  consider  the  ex- 
ceeding difficulty  that  is  found  in  cutting  or  wearing  away  a 
Diamond.  And  a  Circular  motion  of  the  parts  is  much  more 
improbable,  since,  if  that  were  granted,  and  they  be  suppos'd 
irregular  and  Angular  parts,  I  see  not  how  the  parts  of  the 
Diamond  should  hold  so  firmly  together,  or  remain  in  the  same 
sensible  dimensions,  which  yet  they  do.  Next,  if  they  be  Glob- 
ular, and  mov'd  only  with  a  turbinated  motion,  I  know  not  any 
cause  that  can  impress  that  motion  upon  the  pellucid  medium, 
which  yet  is  done.  Thirdly,  any  other  irregular  motion  of  the 

vii 


PREFACE 

parts  one  amongst  another,  must  necessarily  make  the  body 
of  a  fluid  consistence,  from  which-  it  is  far  enough.  It  must 
therefore  be  a  Vibrating  motion. 

"  And  Thirdly,  That  is  a  very  short  vibrating  motion,  I  think 
the  instances  drawn  from  the  shining  of  Diamonds  will  also 
make  probable.  For  a  Diamond  being  the  hardest  body  we  yet 
know  in  the  World,  and  consequently  the  least  apt  to  yield  or 
bend,  must  consequently  also  have  its  vibrations  exceeding 
short. 

"And  these,  I  think,  are  the  three  principal  proprieties  of 
a  motion,  requisite  to  produce  the  effect  call'd  Light  in  the 
Object." — [Micrographia,  pp.  54-56.] 

The  total  absence  of  experimental  evidence  from  the  above 
statement  of  the  case  stands  in  such  marked  contrast  with  the 
method  of  modern  physics  as  initiated  by  Galileo,  that  we  can- 
not for  a  moment  reckon  Hooke  among  the  founders  of  the 
wave-theory. 

So  important,  on  the  contrary,  have  been  the  contributions 
of  Huygens,  Newton,  Young,  and  Fresnel,  that  each  has  in 
turn  been  considered  the  founder  of  the  modern  science  of 
optics.  What  justification  there  is  for  each  of  these  views  will 
be  clearer  from  a  brief  consideration  of  optical  theory  before 
and  after  it  had  been  modified  by  the  work  of  each  of  these 
four  men. 

Two  questions  naturally  arise  in  the  consideration  of  any 
theory,  viz.,  (1)  What  phenomena  does  it  explain? — and  (2) 
How  does  it  explain  them  ?  The  answers  which  have  been 
given  to  these  two  questions  at  various  periods  in  the  develop- 
ment of  the  wave-theory  may  be  outlined  as  follows: 

At  the  time  when  Huygens  and  Newton  began  their  work 
on  light,  the  following  phenomena  were  demanding  explana- 
tion : 

1.  The  existence  of  rays  and  shadows,  known  from  the  ear- 
liest times. 

2.  The  phenomenon  of  reflection,  known  from  the  earliest 
times. 

3.  The  phenomenon  of  refraction,  as  described  by  Snell's  law. 

4.  The  rainbow  and  the  production  of  color  by  the  prism. 

5.  The  colors  of  thin  plates — Newton's  rings. 

6.  Diffraction  bands   outside   the   geometrical  shadow,  de- 
scribed by  Grimaldi,  1665. 


PREFACE 

To  these  might  be  added  the  two  following  phenomena  which 
were  discovered  before  the  final  publication  of  Newton's  Op- 
ticks  (1704)  or  Huygens's  Traite  dela  Lumiere  (1690). 

7.  The  polarization  of  light  by  crystals  (Bartholinus,  1670). 

8.  The  finite  speed  of  light  (Romer,  1675). 

Of  these  eight  cardinal  facts,  the  second,  the  third,  and  the 
eighth,,  were  explained  by  Huygens  on  the  assumption — 

(a)  That  a  luminous  disturbance  consists  of  a  wave-motion 
in  the  ether. 

(b)  That  this  wave-disturbance  travels  with  a  uniform  finite 
speed  through  the  ether  in  any  homogeneous  medium. 

(c)  That  in  different  media  it  travels  with  speeds  which  are 
related  inversely  as  the  refractive  indices  of  those  media. 

But  the  wave-disturbance  as  pictured  by  Huygens  was  a 
single  longitudinal  pulse,  or  blow,  imparted  to  an  elastic  fluid. 
Since  he  did  not  have  in  mind  either  a  train  of  waves  or  trans- 
verse waves,  or  the  idea  of  "  phase,"  or  waves  of  different 
lengths,  it  is  evident  that  he  was  unable  to  explain  any  of  the 
remaining  five  facts. 

Turning  now  to  that  portion  of  the  work  of  Newton  which 
contributed  to  the  wave-theory,  we  find  that  the  fourth  phe- 
nomenon— prismatic  colors — was  explained  by  him  in  1666,  when 
he  demonstrated  that  a  single  ray  of  white  light  contains  all 
the  colors  of  the  spectrum,  and  that  color  is  not  produced  at 
the  surface  of  the  prism,  as  had  been  hitherto  supposed.  This 
discovery  made  possible,  for  the  first  time,  the  correct  explana- 
tion of  the  rainbow. 

In  Newton's  ingenious,  though,  as  we  now  know,  incorrect 
explanation  of  the  fifth  phenomenon — colors  of  thin  plates — we 
meet  the  earliest  measurement  of  the  wave-length  of  light, 
viz.,  the  distance  traversed  by  a  ray  of  light  during  the  inter- 
val between  two  successive  "fits"  of  the  same  kind.  We  meet 
here,  also,  the  first  evidence  that,  in  these  fits,  or,  as  we  now 
say,  waves,  there  is  a  regular  periodicity.  From  this  point  on 
we  must  consider  light  as  travelling  not  only  in  waves,  but  in 
trains  of  waves. 

At  the  close  of  the  period  of  Huygens  and  Newton,  we  have 
then  the  following  facts  still  demanding  explanation  : 

1.  The  existence  of  rays  and  shadows. 

5.  The  colors  of  thin  plates. 
.6.  The  existence  of  diffraction  fringes. 


PREFACE 

7.  The  polarization  of  light  by  crystals. 

To  these  must  now  be  added — 

9.  The  phenomenon  of  stellar  aberration,  discovered  by  Brad- 
ley in  1727. 

Considering  next  the  work  of  Young,  we  find  that  he  first 
suggested  the  correct  explanation  for  the  colors  of  thin  plates, 
having  shown  by  experiment  that  two  rays  of  light  can  inter- 
fere to  produce  alternately  bright  and  dark  bands.  From  this 
experiment  and  the  dark  centre  in  Newton's  rings,  he  con- 
cludes that  light  consists  of  series  of  waves  which,  like  other 
wave  -  motions,  change  phase  by  180°  on  reflection  from  a 
denser  medium. 

Young,  at  this  period  (1802-3),  was  still  laboring  under  the 
impression  that  light-waves  were  longitudinal  and  were  propa- 
gated in  a  fluid  medium  ;  fortunately,  neither  of  these  assump- 
tions affects  the  validity  of  his  reasoning  concerning  the  colors 
of  thin  plates. 

When  Fresnel  began  his  optical  studies  (1814)  the  following 
facts,  viz.,  (1)  existence  of  rays,  (6)  diffraction  fringes,  (7) 
polarization,  and  (9)  aberration,  were  still  to  be  accounted  for 
on  the  wave-theory.  By  the  union  of  Huygens's  principle  with 
the  principle  of  interference,  Fresnel  gave  the  first  satisfac- 
tory explanation  of  the  rectilinear  propagation  of  light,  and 
of  the  existence  of  diffraction  fringes  outside  the  geometrical 
shadow. 

FresneFs  memoir,  in  which  these  discoveries  are  most  sys- 
tematically set  forth,  and  which  was  "  crowned"  by  the  French 
Academy  in  1819,  is  translated  in  the  following  pages.  For 
the  purpose  of  offering  an  elementary  geometrical  explanation 
of  rays  and  diffraction  bands,  Fresnel  invented  the  idea  of 
dividing  the  wave-front  into  a  certain  series  of  zones,  which  in 
nearly  all  text  -  books  are  wrongly  referred  to  as  "  Huygens's 
Zones."  That  this  is  not  only  unfair,  but  also  misleading,  has 
been  pointed  out  by  Professor  Schuster. — Phil.  Mag.  vol.  xxxi., 
p.  77  (1891).  The  first  mention  of  these  Fresnel  Zones,  as 
they  should  be  called,  will  be  found  on  p.  Ill  of  the  present 
volume. 

It  was  in  order  to  explain  the  phenomenon  of  polarization 
that  Fresnel  introduced  the  idea  of  transverse  vibrations  in  the 
ether.  The  boldness  of  this  now  universally  accepted  hypoth- 
esis, which  was  then  practically  equivalent  to  supposing  the 


PREFACE 

ether  an  elastic  solid,  can  be  fully  appreciated  only  after  one 
has  carefully  studied  the  views  of  Fresnel's  contemporaries. 

The  evidence  for  the  transversality  of  light  vibrations  rests 
.pon  the  inability  of  two  oppositely  polarized  rays  to  interfere. 
The  memoir  of  Arago  and  Fresnel  upon  this  subject  is  trans- 
lated in  the  present  volume. 

Of  the  nine  phenomena  which  we  have  more  or  less  arbi- 
trarily selected  as  the  principal  facts  of  optics,  all,  save  only 
the  last  —  aberration  —  had  received  a  fairly  complete  explana- 
tion at  the  close  of  the  labors  of  Young  and  Fresnel.  This 
discovery  of  Bradley's,  which  he  so  easily  disposed  of  on  the 
corpuscular  theory,  has  received  many  explanations  in  terms 
of  the  wave  -theory;  but  none  of  these  can  be  considered  as 
thoroughly  satisfactory.  Young  imagines  the  ether  to  pass 
through  ordinary  matter  "as  freely,  perhaps,  as  the  wind 
passes  through  a  grove  of  trees."  On  this  view,  however,  it 
is  difficult  to  see  how  the  speed  of  light  in  glass,  say,  should 
differ  from  its  speed  in  a  vacuum,  or  how  the  aberration  con- 
stant can  remain  unchanged  when  the  tube  of  the  telescope 
is  filled  with  water,  as  in  Airy's  experiment.  —  Proc.  Roy.  Soc., 
vol.  xx.,  p.  35  (1872). 

For  it  will  be  remembered  that  the  aberration  constant  is 
vl  V  radians,  where 

#=speed  of  earth  in  its  orbit, 

and   F^speed  of  light  between  the  objective  and  eye-piece  of 
telescope  employed. 

Fresnel,  accordingly,  modified  Young's  hypothesis  by  assum- 
ing that,  in  their  motion  through  space,  refracting  bodies  carry 
with  them  only  so  much  ether  as  is  required  to  increase  the 
density  of  free  ether  from  unity  to  p,  where  p  at  any  point  in 
the  medium  is  defined  by  the  following  equation: 


H  being  the  refractive  index  at  the  same  point  in  the  body. 

This  is  really  equivalent  to  saying  that  et  the  luminiferous 
ether  is  entirely  unaffected  by  the  motion  of  the  matter  which 
it  permeates."  [Amer.  Jour.  Sci.,  vol.  cxxxi.,  p.  386.]  And 
that  this  is  the  fact  of  nature  is  exactly  the  conclusion  at  which 
Fizeau  and  Michelson  and  Morley  arrive  from  their  experi- 
ments upon  the  effect  of  motion  of  the  medium  upon  the  speed 
of  light.—  LOG  dt.,  p.  377. 

xi 


PREFACE 

When,  however,  Michelsou  and  Morley  attempt  to  detect 
this  relative  motion  of  the  earth  and  the  ether  as  the  earth 
proceeds  in  its  orbital  motion,  they  do  not  succeed  in  certainly 
finding  that  there  is  any  [Phil.  Mag.,  December,  1887];  and  they 
accordingly  conclude  that  this  relative  motion  is  "quite  small 
enough  to  refute  FresnePs  explanation  of  aberration." 

Of  the  two  experimental  facts  just  cited,  one  apparently 
confirms  FresnePs  view,  and  makes  possible  an  explanation  of 
aberration  in  terms  of  the  wave-theory;  while  the  other  leads 
us  to  think  that  the  ether  moves  with  the  refracting  medium, 
in  which  case  the  wave-theory  appears  incompetent  to  explain 
stellar  aberration. 

It  was  in  the  year  1850  that  Fizeau  and  Foucault  measured 
directly  the  speed  of  light  in  air  and  in  water,  and  found  the 
ratio  of  these  speeds  numerically  equal  to  the  ratio  of  their 
refractive  indices.  This  experiment  has  sometimes  been  called 
the  experimentum  crucis  of  the  wave-theory;  but  with  scant 
justice  we  venture  to  think,  inasmuch  as  no  great  doctrine  in 
physics  can  be  said  to  rest  upon  any  single  fact,  though  mod- 
ification may  be  demanded  by  a  single  fact. 

We  have  now  followed,  in  merest  outline,  the  general  ex- 
planations which  Huygens,  Newton,  Young,  and  Fresnel  have 
offered  for  all,  save  one,  of  this  group  of  nine  cardinal  facts. 
It  is  needless  to  remind  the  reader  that  this  enumeration  forms 
but  a  small  fraction  of  the  phenomena  which  optical  science 
has  brought  to  light  within  the  last  two  centuries,  or,  indeed, 
since  the  labors  of  these  four  men  were  ended. 

No  outline  of  the  wave-theory  would  be  complete  without 
mention  of  the  important  addition  which  was  made  to  it  in 
.the  year  1849  by  Sir  George  Stokes.  For  he  it  was  who  first 
completely  justified  Huygens's  principle  by  showing  that  if  the 
primary  wave  be  resolved  as  proposed  by  Huygens,  no  "back 
wave"  will  be  produced  provided  we  adopt  the  proper  law 
of  disturbance  for  the  secondary  wave.  The  discovery  of  this 
law  was  announced  in  his  memoir  on  the  Dynamical  Theory 
of  Diffraction.  [Trans.  Oamb.  Phil.  Soc.,vo\.  ix.,  p.  1;  Math. 
andPhys.  Papers,  vol.  ii.,  p.  243.]  Mathematically  speaking, 
this  contribution  amounts  to  the  introduction  of  the  factor 
1  -f-  cos  0  into  the  equation  [Eq.  46,  loc.  cU.~\,  which  describes 
the  disturbance  in  a  secondary  wave  proceeding  from  an  ele- 
ment of  the  primary  wave. 

xii 


P  R  E  F  A  C  K 

While,  as  has  been  said  above,  the  following  memoirs  con- 
cern themselves  only  with  the  kinematics  of  light-waves  and 
not  at  all  with  the  question  of  what  is  vibrating,  it  may  not 
be  out  of  place  to  indicate  that  principally  during  the  last  half 
of  the  present  century  at  least  four  more  cardinal  facts  have 
presented  themselves  and  demanded  explanation. 

10.  The  speed  of  light  in  free   space  is  numerically  equal 
to  the  ratio  of  the  electrostatic  and  electromagnetic  units  of 
quantity. 

11.  In  refracting  media,  the  speed  of  light  varies  inversely 
as  the  square  root  of  the  product  of  the  electric  and  magnetic 
inductivities. 

12-.  "  Most  transparent  solid  bodies  are  good  insulators,  and 
all  good  conductors  are  very  opaque." — MAXWELL,  Treatise,  vol. 
ii.,art.  799. 

13.  The  plane  of  polarization  is  rotated  in  a  magnetic  field. 
(Faraday.) 

It  was  to  "  explain"  these  additional  phenomena  that  Max- 
well proposed,  in  1865,  to  modify  the  wave-theory  of  light  by 
replacing  the  mechanical  shear  of  the  ether  by  an  electric  dis- 
placement. How  thoroughly  justified  Maxwell  was  in  this 
move  has  been  amply  proved  mathematically  by  the  analogy 
of  his  equations  with  those  of  the  elastic  solid  theory,  and 
experimentally  by  Hertz  (1888). 

Within  the  last  decade  the  wave -theory  has  shown  itself 
capable  of  explaining  an  entirely  new  group  of  phenomena, 
viz.,  the  color  photography  discovered  by  Lippmann.  Wiener 
has  shown  that  we  have  here  merely  two  rays  of  light — the 
direct  and  reflected  travelling  in  opposite  directions  and  inter- 
fering to  produce  stationary  light  waves. 

The  flexibility  of  the  wave  -  theory  has  still  more  recently 
been  exemplified  by  the  beautiful  discovery  of  Zeeman;  and 
Larmor  and  Preston  have  shown  that  by  assuming  a  particular 
kind  of  electrical  displacement,  viz.,  an  orbital  motion  of  an 
ion,  the  wave-theory  is  competent  to  predict  not  only  the  trip- 
lets and  even  the  sextet,  but  also  the  polarization  produced  by 
placing  the  source  of  radiation  in  a  magnetic  field. — Phil  Mag., 
February,  1899. 

Striking  as  the  resemblance  appears  between  the  kinematics 
of  wave-motion  considered  in  this  volume  and  the  phenomena 
of  optics,  it  must  never  be  forgotten  that  in  all  probability  the 


PREFACE 

vibrating  atom  is  a  structure  whose  motion  is  vastly  compli- 
cated as  compared  with  the  few  simple  motions  which  the' 
experiments  of  Huygens,  Newton,  Young,  Fresnel,  Maxwell, 
and  Michelson  have  assigned  to  it. 

H.  0. 
EVANSTON,  111.,  November,  1899. 

xiv 


GENEBAL   CONTENTS 


PAGE 

Preface v 

Treatise  on  Light.     By  Christiaan  Huygens.    (First  three  chapters).. .       1 

Biographical  Sketch  of  Huygens 42 

On  the  Theory  of  Light  and  Colors.     By  Dr.  Thomas  Young 45 

An  Account  of  Some  Cases  of  the  Production  of  Colors  not  Hitherto 

Described.     By  Dr.  Thomas  Young 62 

Experiments  and  Calculations  relative  to  Physical  Optics.     By  Dr. 

Thomas  Young 68 

Biographical  Sketch  of  Young 77 

Memoir  on  the  Diffraction  of  Light,  crowned  by  the  [French]  Acad- 
emy of  Sciences.    By  A.  J.  Fresnel 79 

On  the  Action  of   Rays   of  Polarized  Light  upon  Each  Other.     By 

Arago  and  Fresnel 145 

Biographical  Sketch  of  Fresnel 156 

Bibliography 161 

Index 165 

xv 


TBEATISE   ON  LIGHT 

CONTAINING 

THE  EXPLANATION  OF  REFLECTION  AND  OF  RE- 
FRACTION AND  ESPECIALLY  OF  THE  RE- 
MARKABLE    REFRACTION    WHICH 
OCCURS  IN  ICELAND  SPAR 

BY 

CHRISTIAAN   HUYGENS 


(Leyden,  1690) 


CONTENTS 

rAGE 

Preface 3 

Table  of  Contents 7 

The  Rectilinear  Propagation  of  Rays  and  some  General  Considerations 

concerning  the  Nature  of  Light 9 

Explanation  of  the  Laws  of  Reflection 25 

Explanation  of  the  Laws  of  Refraction 30 


TREATISE    ON    LIGHT 

BY 

CHRISTIAAN   HUYGENS 


PKEFACE 

THIS  treatise  was  written  during  my  stay  in  Paris  twelve 
years  ago,  and  in  the  year  1678  was  presented  to  the  Eoyal 
Academy  of  Sciences,  to  which  the  king  had. been  pleased  to  call 
me.  Several  of  this  body  who  are  still  living,  especially  those 
who  have  devoted  themselves  to  the  study  of  mathematics, 
will  remember  having  been  at  the  meeting  at  which  I  present- 
ed the  paper;  of  these  I  recall  only  those  distinguished  gentle- 
men Messrs.  Cassini,  Homer,  and  De  la  Hire.  Although  since 
then  I  have  corrected  and  changed  several  passages,  the  copies 
which  I  had  made  at  that  time  will  show  that  I  have  added  noth- 
ing except  some  conjectures  concerning  the  structure  of  Iceland 
spar  and  an  additional  remark  concerning  refraction  in  rock- 
crystal.  I  mention  these  details  to  show  how  long  I  have  been 
thinking  about  these  matters  which  I  am  only  just  now  publish- 
ing, and  not  at  all  to  detract  from  the  merit  of  those  who,  with- 
out having  seen  what  I  have  written,  may  have  investigated 
similar  subjects :  as,  indeed,  happened  in  the  case  of  two  dis- 
tinguished mathematicians.,  Newton  and  Leibnitz,  regarding 
the  question  of  the  proper  figure  for  a  converging  lens,  one 
surface  being  given. 

It  may  be  asked  why  I  have  so  long  delayed  the  publication 
of  this  work.  The  reason  is  that  I  wrote  it  rather  carelessly  in 
French,  expecting  to  translate  it  into  Latin,  and,  in  the  mean- 
time, to  give  the  subject  still  further  attention.  Later  I 

3 


PREFACE 

thought  of  publishing  this  volume  together  with  another  on 
dioptrics  in  which  I  discuss  the  theory  of  the  telescope  and  the 
phenomena  associated  with  it.  But  soon  the  subject  was  no 
longer  new  and  was  therefore  less  interesting.  Accordingly 
I  kept  putting  off  the  work  from  time  to  time,  and  now  I  do 
not  know  when  I  shall  be  able  to  finish  it,  for  my  time  is  large- 
ly occupied  either  by  business  or  by  some  new  investigation. 

In  view  of  these  facts  I  have  thought  wise  to  publish  this 
manuscript  in  its  present  state  rather  than  to  wait  longer  and 
run  the  risk  of  its  being  lost. 

One  finds  in  this  subject  a  kind  of  demonstration  which  does 
not  carry  with  it  so  high  a  degree  of  certainty  as  that  employed 
in  geometry ;  and  which  differs  distinctly  from  the  method 
employed  by  geometers  in  that  they  prove  their  propositions 
by  well-established  and  incontrovertible  principles,  while  here 
principles  are  tested  by  the  inferences  which  are  derivable 
from  them.  The  nature  of  the  subject  permits  of  no  other 
treatment.  It  is  possible,  however,  in  this  way  to  establish  a 
probability  which  is  little  short  of  certainty.  This  is  the  case 
when  the  consequences  of  the  assumed  principles  are  in  perfect 
accord  with  the  observed  phenomena,  and  especially  when 
these  verifications  are  numerous ;  but  above  all  when  one 
employs  the  hypothesis  to  predict  new  phenomena  and  finds 
his  expectations  realized. 

If  in  the  following  treatise  all  these  evidences  of  probability 
are  present,  as,  it  seems  to  me,  they  are,  the  correctness  of  my 
conclusions  will  be  confirmed ;  and,  indeed,  it  is  scarcely  pos- 
sible that  these  matters  differ  very  widely  from  the  picture 
which  I  have  drawn  of  them.  I  venture  to  hope  that  those 
who  enjoy  finding  out  causes  and  who  appreciate  the  wonders 
of  light  will  be  interested  in  these  various  speculations  arid  in 
the  new  explanation  of  that  remarkable  property  upon  which 
the  structure  of  the  human  eye  depends  and  upon  which  are 
based  those  instruments  which  so  powerfully  aid  the  eye.  I 
trust  also  there  will  be  some  who,  -from  such  beginnings,  will 
push  these  investigations  far  in  advance  of  what  I  have  been 
able  to  do ;  for  the  subject  is  not  one  which  is  easily  exhausted. 
This  will  be  evident  especially  from  those  parts  of  the  subject 
which  I  have  indicated  as  too  difficult  for  solution;  and  still 
more  evident  from  those  matters  upon  which  I  have  not 
touched  at  all,  such  as  the  various  kinds  of  luminous  bodies 

4 


PREFACE 

and  the  whole  question  of  color,  which  no  one  can  yet  boast 
of  having  explained. 

Finally,  there  is  much  more  to  be  learned  by  investigation 
concerning  the  nature  of  light  than  I  have  yet  discovered  ;  and 
I  shall  be  greatly  indebted  to  those  who,  in  the  future,  shall 
furnish  what  is  needed  to  complete  my  imperfect  knowledge. 

THE  HAGUE,  8th  of  January,  1690. 


TABLE    OF    CONTENTS 


CHAPTER  I 
ON   THE    RECTILINEAR   PROPAGATION   OF   RAYS 

PAGE 

Light  is  produced  by  a  certain  motion 10 

Particles  do  not  pass  from  the  luminous  object  to  the  eye 10 

Light  is  propagated  radially  very  much  after  the  manner  of  Sound ...  11 

[As  to]  whether  Light  requires  time  for  its  propagation 11 

An  experiment  which  apparently  shows  that  its  transmission  is  in- 
stantaneous    11 

An  experiment  which  shows  that  it  requires  time 13 

Comparison  of  the  Speeds  of  Light  and  Sound 15 

How  the  propagation  of  Light  differs  from  that  of  Sound 15 

They  are  not  each  transmitted  by  the  same  medium. 15 

The  propagation  of  Sound 16 

The  propagation  of  Light 17 

Details  concerning  the  propagation  of  Light 19 

Why  rays  travel  only  in  straight  lines 22 

How  rays  coming  from  different  directions  cross  each  other  without 

interference 23 

CHAPTER   II 
ON    REFLECTION 

Proof  that  the  angles  of  incidence  and  reflection  are  equal  to  each 

other 25 

Why  the  incident  and  reflected  rays  lie  in  one  and  the  same  plane 

perpendicular  to  the  reflecting  surface 27 

Equality  between  the  angles  of  incidence  and  reflection  does  not  de- 
mand that  the  reflecting  surface  be  perfectly  plane 28 

CHAPTER  III 
ON   REFRACTION 

Bodies  may  be  transparent  without  any  matter  passing  through  them  30 

Proof  that  the  ether  can  penetrate  transparent  bodies 31 

How  the  ether  renders  bodies  transparent  by  passing  through  them. .  32 

Bodies,  even  the  most  solid  ones,  have  a  very  porous  structure 32 

7 


TABLE    OF    CONTENTS 

PAGJ? 

The  speed  of  light  is  less  in  water  and  in  glass  than  in  air 32 

A  third   hypothesis  for  the  explanation  of  transparency  and  of  the 

retardation  which  light  undergoes  in  bodies 33 

Concerning  a  possible  cause  of  opacity 34 

Proof  that  refraction  follows  the  Law  of  Sines 34 

Why  the  incident  and  the  refracted  rays  are  each  capable  of  produc- 
ing the  other 35 

Why   reflection    inside  a  triangular  glass   prism   suddenly  increases 

when  the  light  is  no  longer  able  to  emerge 38 

Bodies  in  which  refraction  is  greatest  are  also  those  in  which  reflec- 
tion is  strongest 40 

Demonstration  of  a  theorem  due  to  Fermat . .  40 


CHAPTER  IV 

ON   ATMOSPHERIC    REFRACTION 
[Not  translated.] 

CHAPTER  V 

ON   THE   PECULIAR   REFRACTION    OF   ICELAND    SPAR 
[Not  translated.] 

CHAPTER  VI 

ON   FIGURES   OF   TRANSPARENT   BODIES  ADAPTED   FOR  REFRAC- 
TION   AND    REFLECTION 

[Not  translated.] 

8 


CHAPTER  I 
ON   THE    RECTILINEAR   PROPAGATION   OF   RAYS 

DEMONSTRATIONS  in  optics,  as  in  every  science  where  geome- 
try is  applied  to  matter,  are  based  upon  experimental  facts;  as, 
for  instance,  that  light  travels  in  straight  lines,  that  the  angles 
of  incidence  and  reflection  are  equal,  and  that  rays  of  light  are 
refracted  according  to  the  law  of  sines.  For  this  last  fact  is 
now  as  widely  known  and  as  certainly  known  as  either  of  the 
preceding. 

Most  writers  upon  optical  subjects  have  been  satisfied  to  as- 
sume these  facts.  But  others,  of  a  more  investigating  turn  of 
mind,  have  tried  to  find  the  origin  and  the  cause  of  these 
facts,  considering  them  in  themselves  interesting  natural  phe- 
nomena. And  although  they  have  advanced  some  ingenious 
ideas,  these  are  not  such  that  the  more  intelligent  readers  do 
not  still  want  further  explanation  in  order  to  be  thoroughly 
satisfied. 

Accordingly,  I  here  submit  some  considerations  on  this  sub- 
ject with  the  hope  of  elucidating,  as  best  I  may,  this  depart- 
ment of  natural  science,  which  not  undeservedly  has  gained  the 
reputation  of  being  exceedingly  difficult.  I  feel  myself  espe- 
cially indebted  to  those  who  first  began  to  make  clear  these 
deeply  obscure  matters,  and  to  lead  us  to  hope  that  they  were 
capable  of  simple  explanations. 

But,  on  the  other  hand,  I  have  been  astonished  to  find  these 
same  writers  accepting  arguments  which  are  far  from  evi- 
dent as  if  they  were  conclusive  and  demonstrative.  No  one 
has  yet  given  even  a  probable  explanation  of  the  fundamental 
and  remarkable  phenomena  of  light,  viz*,  why  it  travels  in 
straight  lines  and  how  rays  coining  from  an  infinitude  of  dif- 
ferent directions  cross  one  another  without  disturbing  one  an- 
other. 

I  shall  attempt,  in  this  volume,  to  present  in  accordance  with 

9 


M  KM  01  US    ON 

the  principles  of  modern  philosophy,  some  clearer  ;i,nd  more 
probable  reasons,  first,  for  the  rectilinear  propagation  of  light, 
and,  secondly,  for  its  reflection  when  it  meets  .other  bodies. 
Later  1  shall  explain  the  phenomenon  of  rays  which  art!  said  to 
undergo  refraction  in  passing  through  transparent,  bodies  of 
dilTerenl,  kinds.  Here  1  shall  treat,  also  of  refraction  efl'i-ets  due 
to  the.  varying  density  of  the  earth's  atmosphere.  Afterwards 
I  shall  examine  the  causes  of  that  peculiar  refraction  occur- 
ring in  a  certain  crystal  which  comes  from  Iceland.  And  last  I  y, 
I  shall  consider  the  dill'crenl,  shapes  required  in  transparent  and 
in  rellecting  bodies  to  converge!  rays  upon  a  single  point  or  to 
deflect  them  in  various  ways.  Hero  we  shall  see  with  what 
ease  are  determined,  by  our  new  theory,  not  only  the  ellipses, 
hyperbolas,  and  Other  Curves  which  M.  Descartes  has  so  ingen- 
iously devised  for  this  purpose,  but  also  the  curve  which  one 
surface  of  a  lens  must,  have  when  the  other  surface  is  given,  as 
spherical,  plane,  or  of  any  figure  whatever. 

We  cannot,  help  believing  that  light,  consists  in  the  motion 
of  a  certain  material.  lA>r  when  we  consider  its  production  we 
find  that  here  on  the  earth  it  is  generally  produced  by  fire  and 
llame  which,  beyond  doubt,  contain  bodies  in  a  state  of  rapid 
motion,  since  they  are  able  to  dissolve  and  melt  numerous 
other  more  solid  bodies.  And  if  we  consider  its  effects,  we  see 
that  when  light  is  converged,  as,  for  instance,  by  concave  mir- 
rors, it  is  able  to  produce  combustion  just  as  fire  does  ;  i.e.,  it. 
is  able  to  tear  bodies  apart ;  a  property  that  surely  indicates 
motion,  at  least  in  the  true  philosophy  whore  one  believes  all 
natural  phenomena  to  be  mechanical  effects.  And,  in  my  opin- 
ion, we  must  admit  this,  or  else  give  up  all  hope  of  ever  under- 
standing anything  in  physics. 

Since,  according  to  this  philosophy,  it  is  considered  certain 
thai,  the  sensation  of  sight  is  caused  only  by  the  impulse  of 
some  form  of  matter  upon  tin*  nerves  at  the  base  of  the  eye,  we 
have  here  still  another  reason  for  thinking  that  light  consists 
in  a  motion  of  the  matter  situated  between  us  and  the  lumi- 
nous body. 

When  we  consider,  further,  the  very  great  speed  with  which 
light  is  propagated  in  all  directions,  and  the  fact  that  when 
rays  come  from  different  directions,  even  those  directly  op- 
posite, they  cross  without  disturbing  each  other,  it  must  be 
evident  that  we  do  not  see  luminous  objects  by  means  of  matter 

10 


TIIH    \\AYK-TIIKOHY    OF    LIGHT 

translated  from  the  <>l)jcct  to  us,  as  a  shot  or  MM  arrow  travels 
through  the  air.  For  certainly  this  would  be  in  contradiction 
to  the  two  properties  of  light  which  wo  have  just  mentioned, 
and  especially  to  the  hitter.  Light  is  then  propagated  in  some 
other  manner,  an  understanding  of  which  we  may  obtain  from 
our  knowledge  of  the  manner  in  which  sound  travels  through 
the  air. 

\\  <•  know  that  through  the  medium  of  the  air,  an  invisible 
and  impalpable  body,  sound  is  propagated  in  all  directions, 
from  the  point  where  it  is  produced,  by  means  of  a  motion 
which  is  communicated  successively  from  one  part  of  the  air 
to  another  ;  and  since  this  motion  travels  with  the  same  speed 
in  all  directions,  it  must,  form  spherical  surfaces  which  contin- 
ually enlarge  until  linally  they  strike  our  ear.  Now  there  can 
be  no  doubt  that,  light  also  comes  from  the  luminous  body  to 
us  by  means  of  some  motion  impressed  upon  the  matter  which 
lies  in  the  intervening  space;  for  we  have  already  seen  that 
this  cannot  occur  through  the  translation  of  matter  from  out- 
point to  the  other. 

If.  in  addition,  light  requires  time  for  its  passage — a  point 
we  shall  presently  consider — it  will  then  follow  that  this  motion 
is  impressed  upon  the  matter  gradually,  and  hence  is  propa- 
gated, as  that  of  sound,  by  surfaces  and  spherical  waves.  I 
call  these  'irdrrti  because  of  their  resemblance  to  those 'which 
are  formed  when  one  throws  a  pebble  into  water  and  which 
represent  gradual  propagation  in  circles,  although  produced  by 
a  different  cause  and  confined  <<>  ;t  plane  surface. 

As  to  the  ') nest  ion  of  light  requiring  time  for  its  propaga- 
tion, let  us  consider  first  whether  there  is  any  experimental 
evidence  to  the  contrary. 

What  we  can  do  here  on  the  earth  with  sources  of  light  placed 
at  great,  distanc.es  (alt  hough  showing  that  light  does  not  occupy 
:i  sensible  time  in  passing  over  these  distances)  may  be  objected 
to  on  the  ground  that  these  distances  are  still  too  srnaM,  and 
thai,  therefore,  we  can  conclude  only  that  the  propagation  of 
light  is  exceedingly  rapid.  M.  Descartes  thought  it  instanta- 
neous, and  based  his  opinion  upon  much  better  evidence,  fur- 
nished by  the  eclipse  of  the  moon.  Nevertheless,  as  I  shall 
show,  even  this  evidence  is  not  conclusive.  I  shall  state  the 
matter  in  a  manner  slightly  different  from  his  in  order  that 
we  may  more  easily  arrive  at  all  the  consequences. 

11 


MEMOIRS    ON 


Let  A  be  the  position  of  the  sun ;  BD  a  part  of  the  orbit  or 
annual  path  of  the  earth  ;  ABC  a  straight  line  intersecting  in 
C  the  orbit  of  the  moon,  which  is  represented  by  the  circle  CD. 

If,  now,  light  re- 
quires time  —  say 
one  hour — to  trav- 
erse the  space  be- 
tween the  earth 
and  the  moon,  it 
follows  that  when 
the  earth  has 
reached  the  point 
fig  j  B,  its  shadow,  or 

the     interruption 

of  light,  will  not  yet  have  reached  the  point  C,  and  will  not 
reach  it  until  one  hour  later.  Counting  from  the  time  when 
the  earth  occupies  the  position  B,  it  will  be  one  hour  later  that 
the  moon  arrives  at  the  point  C  and  is  there  obscured  ;  but  this 
eclipse  or  interruption  of  light  will  not  be  visible  at  the  earth 
until  the  end  of  still  another  hour.  Let  us  suppose  that  during 
these  two  hours  the  earth  has  moved  to  the  position  E.  From 
this  point  the  moon  will  appear  to  be  eclipsed  at  C,  a  position 
which  it  occupied  one  hour  before,  while  the  sun  will  be  seen 
at  A.  -For  I  assume  with  Copernicus  that  the  sun  is  fixed  and, 
since  light  travels  in  straight  lines,  must  always  be  seen  in  its 
true  position.  But  it  is  a  matter  of  universal  observation,  we 
are  told,  that  the  eclipsed  moon  appears  in  that  part  of  the 
ecliptic  directly  opposite  the  sun  ;  while  according  to  our  view 
its  position  ought  to  be  behind  this  by  the  angle  GEC,  the 
supplement  of  the  angle  AEC.  But  this  is  contrary  to  the  fact, 
for  the  angle  GEC  will  be  quite  easily  observed,  amounting  to 
about  33°.  Now  according  to  our  computation,  which  will  be 
found  in  the  memoir  on  the  causes  of  the  phenomena  of  Sat- 
urn,-the  distance,  BA,  between  the  earth  and  the  sun  is  about 
12,000  times  the  diameter  of  the  earth,  and  consequently  400 
times  the  distance  of  the  moon,  which  is  30  diameters.  The 
angle  ECB  will,  therefore,  be  almost  400  times  as  great  as 
BAE,  which  is  5',  viz.,  the  angular  distance  traversed  by  the 
earth  in  its  orbit  during  an  interval  of  two  hours.  Thus  the 
angle  BCE  amounts  to  almost  33°,  and  likewise  the  angle 
CEG,  which  is  5'  greater. 

12 


f^" 
( 

THE    WAVE-THEORY    OF    LI'GHT 

^^ 

But  it  must  be  noted  that  in  this  argument  the  speed  of  light 
is  assumed  to  be  such  that  the  time  required  for  it  to  pass  from 
here  to  the  moon  is  one  hour.  If,  however,  we  suppose  that  it 
requires  only  a  minute  of  time,  then  evidently  the  angle  CEG 
will  amount  to  only  33' ;  and  if  it  requires  only  ten  seconds  of 
time,  this  angle  will  amount  to  less  than  6'.  But  so  small  a 
quantity  is  not  easily  observed  in  a  lunar  eclipse,  and  conse- 
quently it  is  not  allowable  to  infer  the  instantaneous  propaga- 
tion of  light. 

It  is  somewhat  unusual,  we  must  'confess,  to  assume  a  speed 
one  hundred  thousand  times  as  great  as  that  of  sound,  which, 
according  to  my  observations,  travels  about  180  toises  [1151 
feet]  in  a  second,  or  during  a  pulse-beat;  but  this  supposition 
appears  by  no  means  impossible,  for  it  is  not  a  question  of  carry- 
ing a  body  with  such  speed,  but  of  a  motion  passing  succes- 
sively from  one  point  to  another. 

I  do  not  therefore,  in  thinking  of  these  matters,  hesitate  to 
suppose  that  the  propagation  of  light  occupies  time,  for  on  this 
view  all  the  phenomena  can  be  explained,  while  on  the  con- 
trary view  none  of  them  can  be  explained.  Indeed,  it  seems  to 
me,  and  to  many  others  also,  that  M.  Descartes,  whose  object 
has  been  to  discuss  all  physical  subjects  in  a  clear  way,  and  who 
has  certainly  succeeded  better  than  any  one,  before  him,  has 
written  nothing  on  light  and  its  properties  which  is  not  either 
full  of  difficulty  or  even  inconceivable. 

But  this  idea  which  I  have  advanced  only  as  a  hypothesis  has 
recently  been  almost  established  as  a  fact  by  the  ingenious 
method  of  Komer,  whose  work  I  propose  here  to  describe,  ex- 
pecting that  he  himself  will  later  give  a  complete  confirmation 
of  this  view. 

His  method,  like  the  one  we  have  just  discussed,  is  astro- 
nomical. He  proves  not  only  that  light  requires  time  for  its 
propagation,  but  shows  also  how  much  time  it  requires  and  that 
its  speed  must  be  at  least  six  times  greater  than  the  estimate 
which  I  have  just  given. 

For  this  demonstration,  he  uses  the  eclipses  of  the  small  plan- 
ets which  revolve  about  Jupiter,  and  which  very  often  pass 
into  its  shadow.  His  reasoning  is  as  follows :  Let  A  denote 
the  sun;  BODE,  the  annual  orbit  of  the  earth;  F,  Jupiter; 
and  GN,  the  orbit  of  the  innermost  satellite,  for  this  one,  on 
account  of  its  short  period,  is  better  adapted  to  this  investi- 

13 


MEMOIRS    ON 


Cation  than  is  either  of  the  other  three.  Let  G  represent  the 
point  of  the  satellite's  entrance  into,  and  H  the  point  of  its 
emergence  from,  Jupiter's  shadow. 

Let  us  suppose  that  an  emergence  of  this 
satellite  has  been  observed  while  the  earth 
occupies  the  position  B,  at  some  time  before 
the  last  quarter.  If  the  earth  remained  in 
this  position,  42J  hours  would  elapse  before 
the  next  emergence  would  occur.  For  this 
is  the  time  required  for  the  satellite  to  make 
one  revolution  in  its  orbit  and  return  to  op- 
position with  the  sun.  If,  for  instance,  the 
earth  remained  at  the  point  B  during  30  rev- 
olutions, then,  after  an  interval  of  30  times 
42-J  hours,  the  satellite  would  again  be  ob- 
served to  emerge.  But  if  meanwhile  the 
earth  has  moved  to  a  point  0,  more  distant 
from  Jupiter,  it  is  evident  that,  provided 
light  requires  time  for  its  propagation,  the 
emergence  of  the  little  planet  will  be  record- 
ed later  at  0  than  it  would  have  been  at  B. 
For  it  will  be  necessary  to  add  to  this  in- 
terval, 30  times  42 \  hours,  the  time  occupied  by  ligh't  in  passing 
over  a  distance  MC,  the  difference  of  the  distances  CH  and  BH. 
In  like  manner,  in  the  other  quarter,  while  the  earth  travels 
from  D  to  E,  approaching  Jupiter,  the  eclipses  will  occur 
earlier  when  the  earth  is  at  E  than  if  it  had  remained  at  D. 

Now  by  means  of  a  large  number  of  these  eclipse  observations, 
covering  a  period  of  ten  years,  it  is  shown  that  these  inequali- 
ties are  very  considerable,  amounting  to  as  much  as  ten  min- 
utes or  more;  whence  it  is  concluded  that,  for  traversing  the 
whole  diameter  of  the  earth's  orbit  KL,  twice  the  distance  from 
here  to  the  sun,  light  requires  about  22  minutes. 

The  motion  of  Jupiter  in  its  orbit,  while  the  earth  passes 
from  B  to  C  or  from  D  to  E,  has  been  taken  into  account  in 
the  computation,  where  it  is  also  shown  that  these  inequalities 
cannot  be  due  either  to  an  irregularity  in  the  motion  of  the 
satellite  or  to  its  eccentricity. 

If  we  consider  the  enormous  size  of  this  diameter,  KL, 
which  I  have  found  to  be  about  24  thousand  times  that  of  the 
earth,  we  get  some  idea  of  the  extraordinary  speed  of  light. 

14 


THE    WAVE-THEORY    OF    LIGHT 

Even  if  we  Suppose  that  KL  were  only  22  thousand  diameters 
of  the  earth,  a  speed  covering  this  distance  in  22  minutes  would 
be  equivalent  to  the  rate  of  one  thousand  diameters  per  minute, 
i.e.,  16f  diameters  a  second  (or  a  pulse-beat),  which  makes  more 
than  eleven  hundred  times  one  hundred  thousand  toises  [212,222 
kilometres],  since  one  terrestrial  diameter  contains  2865  leagues, 
of  which  there  are  £5  to  the  degree,  and  since,  according  to  the 
exact  determination  made  by  Mr.  Picard  in  1669  under  orders 
from  the  king,  each  league  contains  2282  toises. 

But,  as  I  have  said  above,  sound  travels  at  the  rate  of  only 
180  toises  [350  metres]  per  second.  Accordingly,  the  speed  of 
light  is  more  than  600,000  times  as  great  as  that  of  sound, 
which,  however,  is  a  very  different  thing  from  being  instanta- 
neous, the  difference  being  exactly  that  between  a  finite  quantity 
and  infinity.  The  idea  that  luminous  disturbances  are  handed 
on  from  point  to  point  in  a  gradual  manner  being  thus  con- 
firmed, it  follows,  as  I  have  already  said,  that  light  is  propa- 
gated by  spherical  waves,  as  is  the  case  with  sound. 

But  if  they  resemble  each  other  in  this,  respect,  they  differ  in 
several  others — viz.,  in  the  original  production  of  the  motion 
which  causes  them,  in  the  medium  through  which  they  travel, 
and  in  the  manner  in  which  they  are  transmitted  in  this 
medium. 

Sound,  we  know,  is  produced  by  the  rapid  disturbance  of  some 
body  (either  as  a  whole  or  in  part) ;  this  disturbance  setting  in 
motion  the  contiguous  air.  But  luminous  disturbances  must 
arise  at  each  point  of  the  luminous  object,  else  all  the  different 
parts  of  this  object  would  not  be  visible.  This  fact  will  be  more 
evident  in  what  follows. 

In  my  opinion,  this  motion  of  luminous  bodies  cannot  be  bet- 
ter explained  than  by  supposing  that  those  which  are  fluid,  such 
as  a  flame,  and  apparently  the  sun  and  stars,  are  composed 
of  particles  that  float  about  in  a  much  more  subtle  medium, 
which  sets  them  in  rapid  motion,  causing  them  to  strike  against 
the  still  smaller  particles  of  the  surrounding  ether.  But  in  the 
case  of  luminous  solids,  such  as  red-hot  metal  or  carbon,  we 
may  suppose  this  motion  to  be  caused  by  the  violent  disturb- 
ance of  the  particles  of  the  metal  or  of  the  wood,  those  which 
lie  on  the  surface  exciting  the  ether.  Thus  the  motion  which 
produces  light  must  also  be  more  sudden  and  more  rapid  than 
that  which  causes  sound,  since  we  do  not  observe  that  sonorous 

15 


MEMOIRS    ON 

disturbances  give  rise  to  light  any  more  than  that  the  motion 
of  the  hand  through  the  air  gives  rise  to  sound. 

The  question  next  arises  as  to  the  nature  of  the  medium  in 
which  is  propagated  this  motion  produced  by  luminous  bodies. 
I  have  called  it  ether  ;  but  it  is  evidently  something  different 
from  the  medium  through  which  sound  travels.  For  this  lat- 
ter is  simply  the  air  which  we  feel  and  breathe,  and  which, 
when  removed  from  any  region,  leaves  behind  the  luminiferous 
medium.  This  fact  is  shown  by  enclosing  a  sounding  body  in  a 
glass  vessel  and  removing  the  atmosphere  by  means  of  the  air- 
pump  which  Mr.  Boyle  has  devised,  and  with  which  he  has  per- 
formed so  many  beautiful  experiments.  But  in  trying  this  it 
is  well  to  place  the  sounding  body  on  cotton  or  feathers  in  such 
a  way  that  it  cannot  communicate  its  vibrations  either  to  the 
glass  receiver  or  to  the  air-pump,  a  point  which  has  hitherto 
been  neglected.  Then,  when  all  the  air  has  been  removed,  one 
hears  no  sound  from  the  metal  even  when  it  is  struck. 

From  this  we  infer  not  only  that  our  atmosphere,  which  is  un- 
able to  penetrate  glass,  is  the  medium  through  which  sound 
travels,  but  also  that  it  is  different  from  that  which  carries 
luminous  disturbances  ;  for  when  the  vessel  is  exhausted  of 
air,  light  traverses  it  as  freely  as  before. 

This  last  point  is  demonstrated  even  more  clearly  by  the 
celebrated  experiment  of  Torricelli.  That  part  of  the  glass 
tube  which  the  mercury  does  not  fill  contains  a  high  vacuum, 
but  transmits  light  the  same  as  when  filled  with  air.  This 
shows  that  there  is  within  the  tube  some  form  of  matter  which 
is  different  from  air,  and  which  penetrates  either  glass  or  mer- 
cury, or  both,  although  both  the  glass  and  the  mercury  are  im- 
pervious to  air.  And  if  the  same  experiment  is  repeated,  ex- 
cept that  a  little  water  be  placed  on  top  of  the  mercury,  it 
becomes  equally  evident  that  the  form  of  matter  in  question 
passes  either  through  the  glass  or  through  the  water  or  through 
both. 

As  to  the  different  modes  of  transmission  of  Sound  and  light, 
it  is  easy  to  understand  what  happens  in  the  case  of  sound 
when  one  recalls  that  air  can  be  compressed  and  reduced  to 
a  much  smaller  volume  than  it  ordinarily  occupies,  and  that 
just  in  proportion  as  its  volume  is  diminished  it  tends  to  re- 
gain its  original  size.  This  property,  taken  in  conjunction 
with  its  penetrability,  which  it  retains  in  spite  of  compression, 

16 


THE    WAVE-THEORY    OF    LIGHT 

appears  to  show  that  it  is  composed  of  small  particles  which 
float  about,  in  rapid  motion,  in  an  ether  composed  of  still  finer 
particles.  Sound,  then,  is  propagated  by  the  effort  of  these 
air  particles  to  escape  when  at  any  point  in  the  path  of  the  wave 
they  are  more  compressed  than  at  some  other  point. 

But  the  enormous  speed  of  light,  together  with  its  other 
properties,  hardly  allows  us  to  believe  that  it  is  propagated  in 
the  same  way.  Accordingly,  I  propose  to  explain  the  manner 
in  which  I  think  it  must  occur.  It  will  be  necessary  first,  how- 
ever, to  describe  that  property  of  hard  bodies  in  virtue  of  which 
they  transmit  motion  from  one  to  another. 

If  one  takes  a  large  number  of  spheres  of  equal  size,  made  of 
any  hard  material,  and  arranges  them  in  contact  in  a  straight 
line,  he  will  find  that,  on  allowing  a  sphere  of  the  same  size  to 
roll  against  one  end  of  the  line,  the  motion  is  transmitted  in  an 
instant  to  the  other  end  of  the  line*  The  last  sphere  in  the 
row  flies  off  while  the  intermediate  ones  are  apparently  undis- 
turbed ;  the  sphere  which  originally  produced  the  disturbance 
also  remains  at  rest.  Here  we  have  a  motion  which  is  trans- 
mitted with  high  speed,  which  varies  directly  as  the  hardness 
of  the  spheres. 

Nevertheless,  it  is  certain  that  this  motion  is  not  instantane- 
ous, but  is  gradual,  requiring  time.  For  if  the  motion,  or,  if 
you  please,  the  tendency  to  motion,  did  not  pass  successively 
from  one  sphere  to  another,  they  would  all  be  affected  at  the 
same  instant,  and  would  all  move  forward  together.  So  far 
from  this  being  the  case,  it  is  the  last  one  only  which  leaves  the 
row,  and  it  acquires  the  speed  of  the  sphere  which  gave  the  blow. 
Besides  this  experiment  there  are  others  which  show  that  all 
bodies,  even  those  which  are  considered  hardest,  such  as  tem- 
pered steel,  glass,  and  agate,  are  really  elastic,  and  bend  to 
some  extent  whether  they  are  made  into  rods,  spheres,  or  bodies 
of  any  other  shape;  that  is,  they  yield  slightly  at  the  point 
where  they  are  struck,  and  immediately  regain  their  original 
figure.  For  I  have  found  that  in  allowing  a  glass  or  agate 
sphere  to  strike  upon  a  large,  thick,  flat  piece  of  the  same  ma- 
terial, whose  surface  has  been  dulled  by  the  breath,  the  point 
of  contact  is  marked  by  a  circular  disk  which  varies  in  size 
directly  as  the  strength  of  the  blow.  This  shows  that  during 
the  encounter  these  materials  yield  and  then  fly  back,  a  proc- 
ess which  must  require  time. 
B  17 


MEMOIRS    ON 

Now  to  apply  this  kind  of  motion  to  the  explanation  of 
light,  nothing  prevents  our  imagining  the  particles  of  the  ether 
as  endowed  with  a  hardness  almost  perfect  and  with  an  elas- 
ticity as  great  as  we  please.  It  is  not  necessary  here  to  discuss 
the  cause  either  of  this  hardness  or  of  this  elasticity,  for  such  a 
consideration  would  lead  us  too  far  from  the  subject.  I  will, 
however,  remark  in  passing  that  these  ether  particles,  in  spite 
of  their  small  size,  are  in  turn  composed  of  parts,  and  that  their 
elasticity  consists  in  a  very  rapid  motion  of  a  subtle  material 
which  traverses  them  in  all  directions  and  compels  them  to 
assume  a  structure  which  offers  an  easy  and  open  passage  to  this 
fluid.  This  accords  with  the  theory  of  M.  Descartes,  except 
that  I  do  not  agree  with  him  in  assigning  to  the  pores  the 
form  of  hollow  circular  canals.  So  far  from  there  being  any- 
thing absurd  or  impossible  in  all  this,  it  is  quite  credible  that 
nature  employs  an  infinite  series  of  different-sized  molecules, 
endowed  with  different  velocities,  to  produce  her  marvellous 
effects. 

But  although  we  do  not  understand  the  cause  of  elasticity? 
we  cannot  fail  to  observe  that  most  bodies  possess  this  prop- 
erty :  it  is  not  unnatural,  therefore,  to  suppose  that  it  is  a  char- 
acteristic also  of  the  small,  invisible' particles  of  the  ether.  If, 
indeed,  one  looks  for  some  other  mode  of  accounting  for  the 
gradual  propagation  of  light,  he  will  have  difficulty  in  finding 
one  better  adapted  than  elasticity  to  explain  the  fact  of  uniform 
speed.  And  this  appears  to  be  necessary;  for  if  the  motion  slowed 
up  as  it  became  distributed  through  a  larger  mass  of  matter, 
and  receded  farther  from  the  source  of  light,  then  its  high 
speed  would  be  lost  at  great  distances.  But  we  suppose  the 
elasticity  to  be  a  property  of  the  ether  so  that  its  particles  re- 
gain their  shape  with  equal  rapidity  whether  they  are  struck 
with  a  hard  or  a  gentle  blow;  and  thus  the  rate  at  which  the 
light  moves  remains  the  same  [at  all  distances  from  the  source]. 

Nor  is  it  necessary  that  the  ether  particles  should  be  arranged 
in  straight  lines,  as  was  the  ease  with  our  row  of  spheres.  The 
most  irregular  configuration,  provided  the  particles  are  in  con- 
tact with  each  other,  will  not  prevent  their  transmitting  the 
motion  and  handing  it  on  to  the  regions  in  front.  It  is  to  be 
noted  that  we  have  here  a  law  of  motion  which  governs  this 
kind  of  propagation,  and  which  is  verified  by  experiment,  viz., 
when  a  sphere  such  as  A,  touching  several  other  smaller  ones, 

18 


THE    WAVE-THEORY    OF    LIGHT 

CCC,  is  struck  by  another  sphere,  B,  in  such  a  way  as  to  make 
an  impression  upon  each  of  its  neighbors,  it  transfers  its  mo- 
tion to  them  and  remains  at  rest,  as  does  also  the  sphere  B. 
Now,  without  supposing  that  ether  particles  are 
spherical  (for  I  do  not  see  that  this  is  neces- 
sary), we  can  nevertheless  understand  that  this 
Jaw  of  impulses  plays  a  part  in  the  propaga- 
tion of  the  motion. 

Equality  of  size  would  appear  to  be  a  more 
necessary  assumption,  since  otherwise  we  should 
expect  the  motion  to  be  reflected  on  passing 
from  a  smaller  to  a  larger  particle,  following 
the  laws  of  percussion  which  I  published  some  Fig.  3 

years  ago.  Yet,  as  will  appear  later,  this  equal- 
ity is  necessary  not  so  much  to  make  the  propagation  of  light 
possible  as  to  make  it  easy  and  intense.  Nor  does  it  appear 
improbable  that  the  ether  particles  were  made  equal  for  a  pur- 
pose so  important  as  the  transmission  of  light.  This  may  be 
true,  at  least,  in  the  vast  region  lying  beyond  our  atmosphere 
and  serving  only  to  transmit  the  light  of  the  sun  and  the 
stars. 

I  have  now  shown  how  we  may  consider  light  as  propagated, 
in  time,  by  spherical  waves,  arid  how  it  is  possible  that  the 
speed  of  propagation  should  be  as  great  as  that  demanded  by 
experiment  and  by  astronomical  observation.  It  must,  how- 
ever, be  added  that  although  the  ether  particles  are  supposed 
to  be  in  continual  motion  (and  there  is  much  evidence  for  this 
view),  the  gradual  transmission  of  the  waves 
is  not  thus  interfered  with.  For  it  does  not 
consist  in  a  translation  of  these  particles,  but 
merely  in  a  small  vibration,  which  they  are 
compelled  to  transmit  to  their  neighbors  in 
spite  of  their  proper  motion  and  their  change 
of  relative  position. 

But  we  must  consider,  in  greater  detail, 
the  origin  of  these  waves  and  the  manner  of 
their  propagation  from  one  point  to  another. 
And,  first,  it  follows  from  what  has  already 
been  said  concerning  the  production  of  light 
that  each  point  of  a  luminous  body,  such  as 
the  sun,  a  candle,  or  a  piece  of  burning  car- 
19 


MEMOIRS    ON 

bon,  gives  rise  to  its  own  waves,  and  is  the  centre  of  these 
waves.  Thus  if  A,  B,  and  C  represent  different  points  in  a 
candle  flame,  concentric  circles  described  about  each  of  these 
points  will  represent  the  waves  to  which  they  give  rise.  And 
the  same  is  true  for  all  the  points  on  the  surface  and  within 
the  flame.  But  since  the  disturbances  at  the  centre  of  these 
waves  do  not  follow  each  other  in  regular  succession,  we  need 
not  imagine  the  waves  to  follow  one  another  at  equal  intervals; 
and  if,  in  the  figure,  these  waves  are  equally  spaced,  it  is  rather 
to  indicate  the  progress  which  one  and  the  same  wave  has  made 
during  equal  intervals  of  time  than  to  represent  several  waves 
having  their  origin  at  the  same  point.* 

Nor  does  this  enormous  number  of  waves,  crossing  one  an- 
other without  confusion  and  without  disturbing  one  another, 
appear  unreasonable,  for  it  is  well  known  that  one  and  the 
same  particle  of  matter  is  able  to  transmit  several  waves  com- 
ing from  different,  and  even  opposite,  directions.  And  this  is 
true  not  only  in  the  case  where  the  displacements  follow  one 
another  in  succession,  but  also  where  they  are  simultaneous. 
This  is  because  the  motion  is  propagated  gradually.  It  is 
shown  by  the  row  of  hard  and  equal  spheres  above  mentioned. 
If  we  allow  two  equal  spheres,  A  and  D,  to  strike  against  the 
opposite  sides  of  this  row  at  the  same  instant,  they  will  be  ob- 
served to  rebound  each  with  the  same  speed  that  it  had  before 
collision,  while  all  the  other  spheres  remain  at  rest,  although 
the  motion  has  twice  traversed  the  entire  row.  [This  evidently 
implies  that  the  spheres  A  and  D  have  equal  speeds  justi  before 

0    OOO00OO  © 


collision.]  If  these  two  oppositely  directed  motions  happen 
to  meet  at  the  middle  sphere,  B,  or  at  any  other  sphere,  say  0, 
it  will  yield  and  spring  back  from  both  sides,  thus  transmitting 
both  motions  at  the  same  instant. 

*  [From  this  paragraph  it  would  appear  that  Huygens  had  no  conception 
of  trains  of  light-waves.  The  experimental  evidence  for  thinking  that  light- 
waves travel  in  trains  seems  first  to  have  been  furnished  by  Young.  See  pp. 
60,  61  below.  If,  however,  one  prefers  to  interpret  the  colored  rings  of  Newton 
in  terms  of  the  wave-theory,  this  experimental  evidence  may  be  ascribed  to 
Newton.'] 

30 


THE    WAVE-THEORY    OF    LIGHT 

Bnt  what  is  strangest  and  most  astonishing  of  all  is  that  waves 
produced  by  displacements  and  particles  so  minute  should  spread 
to  distances  so  immense,  as,  for  instance,  from  the  sun  or  from 
the  stars  to  the  earth.  For  the  intensity  of  these  waves  must 
diminish  in  proportion  to  their  distance  from  the  origin  until 
finally  each  individual  wave  is  of  itself  unable  to  produce  the 
sensation  of  light.  Our  astonishment,  however,  diminishes 
when  we  consider  that  in  the  great  distance  which  separates 
us  from  the  luminous  body  there  is  an  infinitude  of  waves 
which,  although  coming  from  different  parts  of  the  [luminous] 
body,  are  practically  compounded  into  a  single  wave  which  thus 
acquires  sufficient  intensity  to  affect  our  senses.  Thus  the  in- 
finitely great  number  of  waves  which  at  any  one  instant  leave 
a  fixed  star,  as  large  possibly  as  our  sun,  unite  to  form  what 
is  equivalent  to  one  single  wave*  of  intensity  sufficient  to  affect 
the  eye.  Not  only  so,  but  each  luminous  point  may  send  us 
thousands  of  waves  in  the  shortest  imaginable  time,  on  account 
of  r!  e  rapidity  of  the  blows  with  which  the  particles  of  the 
luminous  body  strike  the  ether  at  these  points.  The  effect  of 
the  waves  would  thus  be  rendered  still  more  sensible. 

In  considering  the  propagation  of  waves,  we  must  remember 
that  each  particle  of  the  medium  through  which  the  wave 
spreads  doe  not  communicate  its  motion  only  to  that  neighbor 
which  lies  in  the  straight  line  drawn  from  the  luminous  point, 
but  shares  it  with  all  the  particles  which  touch  it  and  resist  its 
motion.  Each  particle  is  thus  to  be  considered  as  the  centre 
of  a  wave.  Thus  if  DCF  is  a  wave  whose  centre  and  origin 
is  the  luminous  point  A,  a  parti- 
cle at  B,  inside  the  sphere  DCF, 
will  give  rise  to  its  own  individual 
[secondary]  wave,  KCL,  which  will 
touch  the  wave  DCF  in  the  point 
C,  at  the  same  instant  in  which  the 
principal  wave,  originating  at  A, 
reaches  the  position  DCF.  And  it 
is  clear  that  there  will  be  only  one  D 
point  of  the  wave  KCL  which  will 
touch  the  wave  DCF,  viz.,  the  point  Fig.  6 

which  lies  in  the  straight  line  from  A 

drawn  through  B.     In  like  manner,  each  of  the  other  particles, 
bbbb,  etc.,   lying   within   the   sphere   DCF,   gives   rise   to   its 

21 


MEMOIRS    ON 


own  wave.  The  intensity  of  each  of  these  waves  may,  how- 
ever, be  infinitesimal  compared  with  that  of  DCF,  which  is 
the  resultant  of  all  those  parts  of  the  other  waves  which  are  at 
a  maximum  distance  from  the  centre  A. 

We  see,  moreover,  that  the  wave  DCF  is  determined  by  the 
extreme  limit  to  which  the  motion  has  travelled  from  the  point 
A  within  a  certain  interval  of  time.  For  there  is  no  motion 
beyond  this  wave,  whatever  may  have  been  produced  inside  by 
those  parts  of  the  secondary  waves  which  do  not  touch  the 
sphere  DCF.  Let  no  one  think  this  discussion  mere  hair- 
splitting. For,  as  the  sequel  will  show,  this  principle,  so  far 
from  being  an  ultra-refinement,  is  the  chief  element  in  the  ex- 
planation of  all  the  properties  of  light,  including  the  phe- 
nomena of  reflection  and  refraction.  This  is  exactly  the  point 
which  seems  to  have  escaped  the  attention  of  those  who  first 
took  up  the  study  of  light-waves,  among  whom  are  Mr.  Hooke, 
in  his  Miorographia,  and  Father  Pardies,  who  had  undertaken 
to  explain  reflection  and  refraction  on  the  wave -theory,  as  I 
know  from  his  having  shown  me  a  part  of  a  memoir  which  he 
was  unable  to  finish  before  his  death.  But  the  most  important 
fundamental  idea,  which  consists  in  the  principle  I  have  just 
stated,  is  wanting  in  his  demonstrations.  On  other  points  also 
his  view  is  different  from  mine,  as  will  some  day  appear  in  case 
his  writings  have  been  preserved. 

Passing  now  to  the  properties  of  light,  we  observe  first  that 
each  part  of  the  wave  is  propagated  in  such  a  way  that  its  ex- 
tremities lie  always  between  the  same 
straight  lines  drawn  from  the  lumi- 
nous point. 

For  instance,  that  part  of  the  wave 
BGr,  whose  centre  is  the  luminous 
point  A,  develops  into  the  arc  CE, 
limited  by  the  straight  lines,  ABO 
and  AGE.  For  although  the  sec- 
ondary waves  produced  by  the  par- 
ticles lying  within  the  space  CAE 
may  spread  to  the  region  outside, 
nevertheless  they  do  not  combine  at 
the  same  instant  to  produce  one  single  wave  limiting  the 
motion  and  lying  in  the  circumference  CE  which  is  their 
common  tangent.  This  explains  the  fact  that  light,  pro- 

22 


Rff.fi 


THE    WAVE-THEORY    OF    LIGHT 

vided  its  rays  are  not  reflected  or  refracted,  always  travels 
in  straight  lines,  so  that  no  body  is  illuminated  by  it  unless 
the  straight -line  path  from  the  source  to  this  body  is  unob- 
structed. 

Let  us,  for  instance,  consider  the  aperture  BG-  as  limited  by 
the  opaque  bodies  BH,  GI ;  then,  as  we  have  just  indicated, 
the  light- waves  will  always  be  limited  by  the  straight  lines 
AC,  AE.  The  secondary  waves  which  spread  into  the  region 
outside  of  ACE  have  not  sufficient  intensity  to  produce  the 
sensation  of  light. 

Now,  however  small  we  may  make  the  opening  BG,  the  cir- 
cumstances which  compel  the  light  to  travel  in  straight  lines 
still  remain  the  same ;  for  this  aperture  is  always  sufficiently 
large  to  contain  a  great  number  of  these  exceedingly  minute 
ether  particles.  It  is  thus  evident  that  each  particular  part  of 
any  wave  can  advance  only  along  the  straight  line  drawn  to  it 
from  the  luminous  point.  And  this  justifies  us  in  considering 
rays  of  light  as  straight  lines. 

From  what  has  been  said  concerning  the  small  intensity  of 
the  secondary  waves,  it  would  appear  not  to  be  necessary  that 
all  the  ether  particles  be  equal,  although  such  an  equality 
would  favor  the  propagation  of  the  motion.  The  effect  of 
inequality  would  be  to  make  a  particle,  in  colliding  with  a 
larger  one,  use  up  a  part  of  its  momentum  in  an  effort  to 
recover.  The  secondary  waves  thus  sent  backward  towards 
the  luminous  point  would  be  unable  to  produce  the  sensation 
of  light,  and  would  not  result  in  a  primary  wave  similar  to 
CE. 

Another  and  more  remarkable  property  of  light  is  that 
when  rays  come  from  different,  or  even  opposite,  directions 
each  produces  its  effect  without  disturbance  from  the  other. 
Thus  several  observers  are  able,  all  at  the  same  time,  to 
look  at  different  objects  through  one  single  opening ;  and 
two  individuals  can  look  into  each  other's  eyes  at  the  same 
instant. 

If  we  now  recall  our  explanation  of  the  action  of  light  and  of 
waves  crossing  without  destroying  or  interrupting  each  other, 
these  effects  which  we  have  just  described  are  readily  under- 
stood, though  they  are  not  so  easily  explained  from  Descartes' 
point  of  view,  viz.,  that  light  consists  in  a  continuous  [hydro- 
static] pressure  which  produces  only  a  tendency  to  motion. 

23 


MEMOIRS    ON    THE    WAVE-THEORY    OF    LIGHT 

For  such  a  pressure  cannot,  at  the  same  instant,  affect  bodies 
from  two  opposite  sides  unless  these  bodies  have  some  tendency 
to  approach  each  other.  It  is,  therefore,  impossible  to  under- 
stand how  two  persons  can  look  each  other  in  the  eye  or  how 
one  torch  can  illuminate  another. 

24 


CHAPTER  II 


ON     REFLECTION 

HAVING  explained  the  effects  produced  by  light-waves  in  a 
homogeneous  medium,  we  shall  next  consider  what  happens 
when  they  impinge  upon  other  bodies.  First  of  all  we  shall 
see  how  reflection  is  explained  by  these  same  waves  and  how 
the  equality  of  angles  fol- 
lows as  a  consequence. 
Let  AB  represent  a  plane 
polished  surface  of  some 
metal,  glass,  or  other  sub- 
stance, which,  for  the  pres- 
ent, we  shall  consider  as 
perfectly  smooth  (concern- 
ing irregularities  which 
are  unavoidable  we  shall 
have  something  to  say  at 
the  close  of  this  demon- 
stration) ;  and  let  the  line 
AC,  inclined  to  AB,  repre- 
sent a  part  of  a  light-wave  whose  centre  is  so  far  away  that  this 
part  AC  may  be  considered  as  a  straight  line.  It  may  be  men- 
tioned here,  once  for  all,  that  we  shall  limit  our  consideration 
to  a  single  plane,  viz.,  the  plane  of  the  figure,  which  passes 
through  the  centre  of  the  spherical  wave  and  cuts  the  plane 
AB  at  right  angles. 

The  region  immediately  about  C  on  the  wave  AC  will,  after 
a  certain  interval  of  time,  reach  the  point  B  in  the  plane  AB, 
travelling  along  the  straight  line  CB,  which  we  may  think  of 
as  drawn  from  the  source  of  light  and  hence  drawn  perpen- 
dicular to  AC.  Now  in  this  same  interval  of  time  the  ^egion 
about  A  on  the  same  wave  is  unable  to  transmit  its  entire 
motion  beyond  the  plane  AB  ;  it  must,  therefore,  continue  its 

25 


MEMOIRS    ON 


motion  on  this  side  of  the  plane  to  a  distance  equal  to  CB, 
sending  out  a  secondary  spherical  wave  in  the  manner  described 
above.  This  secondary  wave  is  here  represented  by  the  circle 
SNR,  drawn  with  its  centre  at  A  and  with  its  radius  AN  equal 
to  CB. 

So,  also,  if  we  consider  in  turn  the  remaining  parts  H  of  the 
wave  AC,  it  will  be  seen  that  they  not  only  reach  the  surface 
AB  along  the  straight  lines  HK  parallel  to  CB,  but  they  will 
produce,  at  the  centres  K,  their  own  spherical  waves  in  the 
transparent  medium.  These  secondary  waves  are  here  repre- 
sented by  circles  whose  radii  are  equal  to  KM — that  is,  equal 
to  the  prolongations  of  HK  to  the  straight  line  BG  which 
is  drawn  parallel  to  AC.  But,  as  is  easily  seen,  all  these  cir- 
cles have  a  common  tangent  in  the  straight  line  BN,  viz., 
the  same  line  which  passes  through  B  and  is  tangent  to 
the  first  circle  having  A  as  centre  and  AN,  equal  to  BC,  as 
radius. 

This  line  BN  (lying  between  B  and  the  point  N,  the  foot  of 
the  perpendicular  let  fall  from  A)  is  the  envelope  of  all  these 
circles,  and  marks  the  limit  of  the  motion  produced  by  the 
reflection  of  the  wave  AC.  It  is  here  that  the  motion  is  more 

intense  than  at  any  other 
point,  because,  as  has  been 
explained,  BN  is  the  new 
position  which  the  wave 
AC  has  assumed  at  the  in- 
stant when  the  point  C  has 
reached  B.  For  there  is 
no  other  line  which,  like 
BN,  is  a  common  tangent 
to  these  circles,  unless  it 
be  BG,  on  the  other  side 
of  the  plane  AB.  And  BGr 
will  represent  the  trans- 
mitted wave  onlv  in  case 


Fig.  7 


the  motion  occurs  in  a  medium  which  is  homogeneous  with 
that  above  the  plane.  If,  however,  one  wishes  to  see  just  how 
the  wave  AC  has  gradually  passed  into  the  wave  BN,  he  has 
only  to  use  the  same  figure  and  draw  the  straight  lines  KO 
parallel  to  BN,  and  the  straight  lines  KL  parallel  to  AC.  It  is 
thus  seen  that  the  wave  AC,  from  being  a  straight  line,  passes 

26 


THE    WAVE-THEORY    OF    LIGH 


successively  into  all  the  broken  lines  OKL,  and  reassumes  the 
form  of  a  single  straight  line  NB. 

It  is  now  evident  that  the  angle  of  reflection  is  equal  to  the 
angle  of  incidence.  For  the  right-angled  triangles  ABC  and  BNA 
have  the  side  AB  in  common,  and  the  side  OB  equal  to  the  side 
NA,  whence  it  follows  that  the  angles  opposite  these  sides  are 
equal,  and  hence  also  the  angles  CBA  and  NAB.  But  .CB, 
perpendicular  to  CA,  is  the  direction  of  the  incident  ray,  while 
AN,  perpendicular  to  the  wave  BN,  has  the  direction  of  the 
reflected  ray.  These  rays  are,  therefore,  equally  inclined  to  the 
plane  AB. 

Against  this  demonstration  it  may  he  urged  that  while  BN 
is  the  common  tangent  of  the  circular  waves  in  the  plane  of 
this  figure,  the  fact  is  that  these  waves  are  really  spherical  and 
have  an  infinitely  great  number  of  similar  tangents,  viz.,  all 
straight  lines  drawn  through  the  point  B  and  lying  in  the  sur- 
face of  a  cone  generated  by  the  revolution  of  a  straight  line 
BN  about  BA  as  axis.  It  remains  to  be  shown,  therefore,  that 
this  objection  presents  no  difficulty  ;  and,  incidentally,  we  shall 
see  that  the  incident  and  reflected  rays  each  lie  in  one  plane 
perpendicular  to  the  reflecting  plane. 

I  remark,  then,  that  the  wave  AC,  so  long  as  it  is  considered 
merely  a  line,  can  produce  no  light.  For  a  ray  of  light,  how- 
ever slender,  must  have  a  finite  thickness  in  order  to  be  visible. 
In  order,  therefore,  to  represent  a  wave  whose  path  is  along 
this  ray,  it  is  necessary  to  replace  the  straight  line  AC  by  a 
plane  area,  as,  for  instance,  by  the  circle  HC  in  the  following 
figure,  where  the  luminous  point  is  supposed  to  be  infinitely 
distant.  From  the  preceding  proof  it  is  easily  seen  that  each 
element  of  area  on  the  wave  HC,  having  reached  the  plane  AB, 
will  there  give  rise  to  its  own  secondary  wave ;  and  when  C 
reaches  the  point  B,  these  will  all  have  a  common  tangent 
plane,  viz..  the  circle  BN  equal  to  CH.  This  circle  will  be  cut 
through  the  centre  and  at  right  angles  by  the  same  plane  which 
thus  cuts  the  circle  CH  and  the  ellipse  AB. 

It  is  thus  seen  that  the  spherical  secondary  waves  can  have 
no  common  tangent  plane  other  than  BN.  In  this  plane  will 
be  located  more  of  the  reflected  motion  than  in  any  other,  and 
it  will  therefore  receive  the  light  transmitted  from  the  wave  CH. 
I  have  noted  in  the  preceding  explanation  that  the  motion  of 
the  wave  incident  at  A  is  not  transmitted  beyond  the  plane  AB, 

27 


MEMOIRS    ON 

at  least  not  entirely.  And  here  it  is  necessary  to  remark  that, 
although  the  motion  of.  the  ether  may  be  partly  communicated 
to  the  reflecting  body,  this  cannot  in  the  slightest  alter  the 
speed  of  the  propagation  of  the  waves,  which  determines  the 
angle  of  reflection.  For,  in  any  one  medium,  a  slight  disturb- 
ance produces  waves  which  travel  with  the  same  speed  as  those 


Fig.  8 

due  to  a  very  great  disturbance,  a  consequence  of  that  property 
of  elastic  bodies  concerning  which  we  have  spoken  above,  viz., 
the  time  occupied  in  recovery  is  the  same  whether  the  com- 
pression be  large  or  small.  In  every  case  of  reflection  of  light 
from  the  surface  of  any  substance  whatever  the  angles  of  in- 
cidence and  reflection  are  therefore  equal,  even  though  the 
body  be  of  such  a  nature  as  to  absorb  a  part  of  the  motion  de- 
livered .by  the  incident  wave.  And,  indeed,  experiment  shows 
that  among  polished  bodies  there  is  no  exception  to  this  law 
of  reflection. 

"We  must  emphasize  the  fact  that  in  our  demonstration  there 
is  no  need  that  the  reflecting  surface  be  considered  a  perfectly 
smooth  plane,  as  has  been  assumed  by  all  those  who  have  at- 
tempted to  explain  the  phenomena  of  reflection.  All  that  is 
called  for  is  a  degree  of  smoothness  such  as  would  be  produced 
by  the  particles  of  the  reflecting  medium  being  placed  one  near 
another.  These  particles  are  much  larger  than  those  of  the 
ether,  as  will  be  shown  later  when  we  come  to  treat  of  the 
transparency  and  opacity  of  bodies.  Since,  now,  the  surface 
consists  thus  of  particles  assembled  together,  the  ether  par- 
ticles being  above  and  smaller,  it  is  evident  that  one  cannot 
demonstrate  the  equality  of  the  angles  of  incidence  and  reflec- 
tion from  the  time-worn  analogy  with  that  which  happens  when 

28 


THE    WAVE-THEORY    OF    LIGHT 

a  ball  is  thrown  against  a  wall.     By  our  method,  on  the  other 
hand,  the  fact  is  explained  without  difficulty. 

Take  particles  of  mercury,  for  instance,  for  they  are  so 
small  that  we  must  think  of  the  least  visible  portion  of  surface 
as  containing  millions,  arranged  like  the  grains  in  a  heap  of 
sand  which  one  has  smoothed  out  as  much  as  possible;  this 
surface  for  our  purpose  is  equal  to  polished  glass.  And,  though 
such  a  surface  may  be  always  rough  compared  with  ether  par- 
ticles, it  is  evident  that  the  centres  of  all  the  secondary  waves 
of  reflection  which  we  have  described  above  lie  practically  in 
one  plane.  Accordingly,  a  single  tangent  comes  as  near  touch- 
ing them  all  as  is  necessary  for  the  production  of  light.  And 
this  is  all  that  is  required  in  our  demonstration  to  explain  the 
equality  of  angles  without  allowing  the  rest  of  the  motion,  re- 
flected in  various  directions,  to  produce  any  disturbing  elfect. 


CHAPTER  III 
OK     REFRACTION 

IK  the  same  manner  that  reflection  has  been  explained  by 
light-waves  reflected  at  the  surface  of  polished  bodies,  we  pro- 
pose now  to  explain  transparency  and  the  phenomena  of  refrac- 
tion by  means  of  waves  propagated  into  and  through  transpar- 
ent bodies,  whether  solids,  such  as  glass,  or  liquids,  such  as 
water  and  oils.  But,  lest  the  passage  of  waves  into  these 
bodies  appear  an  unwarranted  assumption,  I  will  first  show  that 
this  is  possible  in  more  ways  than  one. 

Let  us  imagine  that  the  ether  does  penetrate  any  transparent 
body,  its  particles  will  still  be  able  to  transmit  the  motion  of 
the  waves  just  as  do  those  of  the  ether,  supposing  them  each  to 
be  elastic.  And  this  we  can  easily  believe  to  be  the  case  with 
water  and  other  transparent  liquids,  since  they  are  composed 
of  discrete  particles.  But  it  may  appear  more  difficult  in  the 
case  of  glass  and  other  bodies  that  are  transparent  and  hard, 
because  their  solidity  would  hardly  allow  that  they  should  take 
up  any  motion  except  that  of  their  mass  as  a  whole.  This, 
however,  is  not  necessary,  since  this  solidity  is  not  what  it  ap- 
pears to  us  to  be,  for  it  is  more  probable  that  these  bodies  are 
composed  of  particles  which  are  placed  near  one  another  and 
bound  together  by  an  external  pressure  due  to  some  other  kind 
of  matter  and  by  irregularity  of  their  own  configurations.  For 
their  looseness  of  structure  is  seen  in  the  facility  with  which 
they  are  penetrated  by  the  medium  of  magnetic  vortices  and 
those  which  cause  gravitation.  One  cannot  go  further  than  to 
say  that  these  bodies  have  a  structure  similar  to  that  of  a  sponge, 
or  of  light  bread,  because  heat  will  melt  them  and  change  the 
relative  positions  of  their  particles.  We  infer,  then,  as  has 
been  indicated  above,  that  they  are  assemblages  of  particles 
touching  one  another  but  not  forming  a  continuous'  solid. 
This  being  the  case,  the  motion  which  these  particles  receive 

30 


MEMOIRS    ON    THE   WAVE-THEORY    OF    LIGHT 

in  the  transmission  of  light  is  simply  communicated  from  one 
to  another,  while  the  particles  themselves  remain  tethered  in 
their  own  positions  and  do  not  become  disarranged  among 
themselves.  It  is  easily  possible  for  this  to  occur  without  in 
any  way  affecting  the  solidity  of  the  structure  as  seen  by  us. 

By  the  external  pressure  of  which  I  have  spoken  is  not  to  be 
understood  that  of -'the  air,  which  would  be  quite  insufficient, 
but  that  of  another  and  more  subtle  medium,  whose  pressure  is 
exhibited  by  an  experiment  which  I  chanced  upon  a  long  while 
ago,  namely,  that  water  from  which  the  air  has  been  removed 
remains  suspended  in  a  glass  tube  open  at  the  lower  end,  even 
though  the  air  may  have  been  exhausted  from  a  vessel  enclosing 
this  tube. 

We  may  thus  explain  transparency  without  assuming  that 
bodies  are  penetrated  by  the  luminiferous  ether  or  that  they 
contain  pores  through  which  the  ether  can  pass.  The  fact, 
however,  is  not  only  that  this  medium  penetrates  ordinary 
bodies,  but  that  it  does  so  with  the  utmost  ease,  as  indeed  has 
already  been  shown  by  the  experiment  of  Torricelli  which  we 
have  cited  above.  When  the  mercury  or  the  water  leaves  the 
upper  part  of  the  glass  tube,  the  ether  appears  at  once  to  take 
its  place  and  transmit  light.  But  following  is  still  another 
argument  for  thinking  that  bodies,  not  only  those  which  are 
transparent,  but  others  also,  are  easily  penetrable. 

When  light  traverses  a  hollow  glass  sphere  which  is  com- 
pletely closed,  it  is  evident  that  the  sphere  is  filled  with  ether, 
just  as  is  the  space  outside  the  sphere.  And  this  ether,  as  we 
have  shown  above,  consists  of  particles  lying  in  close  contact 
with  each  other.  If,  now,  it  were  enclosed  in  the  sphere  in  such 
a  way  that  it  could  not  escape  through  the  pores  of  the  glass, 
it  would  be  compelled  to  partake  of  any  motion  which  one 
might  impress  upon  the  sphere  ;  ^consequently  nearly  the  same 
force  would  be  required  to  impress  a  given  speed  upon  this 
sphere,  lying  upon  a  horizontal  plane,  as  if  it  were  filled  with 
water,  or  possibly  mercury.  For  the  resistance  which  a  body 
offers  to  any  velocity  one  may  wish  to  impart  to  it  varies 
directly  as  the  quantity  of  matter  which  the  body  contains  and 
which  is  compelled  to  acquire  velocity.  But  the  fact  is  that 
the  sphere  resists  the  motion  only  in  proportion  to  the  amount 
of  glass  in  it.  Whence  it  follows  that  the  ether  within  is  not 
enclosed,  but  flows  through  the  glass  with  perfect  freedom. 

31 


MEMOIRS    ON 

Later  we  shall  show,  by  this  same  process,  that  penetrability 
may  be  inferred  for  opaque  bodies  also. 

A  second  and  more  probable  explanation  of  transparency 
is  to  say  that  the  light-waves  continue  on  in  the  ether  which 
always  fills  the  interstices,  or  pores,  of  transparent  bodies. 
For  since  it  passes  continuously  and  with  ease,  it  follows  that 
these  pores  are  always  full.  Indeed,  it  may  be  shown  that  these 
interstices  occupy  more  space  than  the  particles  which  make 
up  the  body. 

Now  if  it  be  true,  as  we  have  said,  that  the  force  required 
to  impart  a  given  horizontal  velocity  to  a  body  is  proportional 
to  the  mass  of  the  body,  and  if  this  force  be  also  proportional 
to  the  weight  of  the  body,  as  we  know  by  experiment  that  it 
is,  then  the  mass  of  any  body  must  be  ^also  proportional  to 
its  weight.  Now  we  know  that  water  weighs  only  -fa  part  as 
much  as  an  equal  volume  of  mercury,  therefore  the  substance 
of  the  water  occupies  only  -fa  part  of  the  space  that  encloses 
its  mass.  Indeed,  it  must  occupy  even  a  smaller  fraction  than 
this,  "because  mercury  is  not  so  heavy  as  gold,  and  gold  is  a 
substance  which  is  not  very  dense,  since  the  medium  of  mag- 
netic vortices  and  that  which  causes  gravitation  penetrate  it 
with  the  utmost  ease. 

But  it  may  be  objected  that  if  water  be  a  substance  of  such 
small  density,  and  if  its  particles  occupy  so  small  a  portion  of 
its  apparent  volume,  it  is  very  remarkable  that  it  should  offer 
such  stubborn  resistance  to  compression  ;  for  it  has  not  been 
condensed  by  any  force  hitherto  employed,  and  remains  per- 
fectly liquid  while  under  pressure. 

This  is,  indeed,  no  small  difficulty.  But  it  may  nevertheless 
be  explained  by  supposing  that  the  very  violent  and  rapid 
motion  of  the  subtle  medium  which  keeps  water  liquid  also 
sets  in  motion  the  particles  of  which  it  is  composed,  and  main- 
tains this  liquid  state  in  spite  of  any  pressure  which  has  hitherto 
been  applied. 

If,  now,  the  structure  of  transparent  bodies  be  as  loose  as  we 
have  indicated,  we  may  easily  imagine  waves  penetrating  the 
ether  which  fills  the  interstices  between  the  particles.  Not 
only  so,  but  we  can  easily  believe  that  the  speed  of  these  waves 
inside  the  body  must  be  a  little  less  on  account  of  the  small 
detours  necessitated  by  these  same  particles.  I  propose  to  show 
that  in  this  varying  velocity  of  light  lies  the  cause  of  refraction. 

32 


THE    WAVE-THEORY    OF    LIGHT 

I  will  first  indicate  a  third  and  last  method  by  which  we  may 
explain  transparency,  namely,  by  supposing  that  the  motion  of 
the  light-waves  is  transmitted  equally  well  by  the  ether  particles 
which  fill  the  interstices  of  the  body,  and  by  the  particles  which 
compose  the  body,  the  motion  being  handed  on  from  one  to  the 
other.  A  little  later  we  shall  see  how  beautifully  this  hypothesis 
explains  the  double'refraction  of  certain  transparent  substances. 
Should  one  object  that  the  particles  of  ether  are  much  smaller 
than  those  of  the  transparent  body,  since  the  former  pass 
through  the  intervals  between  the  latter,  and  that  consequent- 
ly they  would  be  able  to  communicate  only  a  small  portion  of 
their  momentum,  we  may  reply  that  the  particles  of  the  body 
are  composed  of  other  still  smaller  particles,  and  that  it  is  these 
secondary  particles  that  take  up  the  momentum  from  those  of 
the  ether. 

Moreover,  if  the  particles-  of  the  transparent  body  are  slight- 
ly less  elastic  than  are  the  ether  particles,  which  we  may 
reasonably  suppose,  it  would  still  follow  that  the  speed  of 
the  light  waves  is  less  inside  the  body  than  outside  in  the 
ether. 

We  have  here,  what  appears  to  me,  the  manner  in  which 
light-waves  are  probably  transmitted  by  transparent  bodies. 
But  there  still  remains  the  consideration  of  opaque  bodies  and 
the  difference  between  these  and  transparent  bodies,  a  question 
all  the  more  interesting  in  view  of  the  ease  with  which  ether 
penetrates  all  bodies,  a  fact  to  which  attention  has  already  been 
directed,  and  which  might  lead  one  to  think  that  all  bodies 
should  be  transparent.  For  considering  the  hollow  sphere,  by 
which  I  have  shown  the  open  structure  of  glass  and  the  ease 
with  which  ether  passes  through  it,  one  would  naturally  infer 
the  same  penetrability  as  a  property  of  metals  and  all  other 
substances.  Imagine  the  sphere  to  be  of  silver;  it  would  then 
certainly  contain  luminiferous  ether,  because  this  substance, 
as  well  as  air,  would  be  present  in  it  when  the  opening  in  the 
sphere  was  closed  up.  But  when  closed  and  placed  upon  a 
horizontal  plane  it  would  resist  motion  only  in  proportion  to 
the  amount  of  silver  in  it,  showing  as  above  that  the  enclosed 
ether  does  not  acquire  the  motion  of  the  sphere.  Silver,  there- 
fore, like  glass,  is  easily  penetrated  by  ether.  In  between  the" 
particles  of  silver  and  of  all  other  opaque  bodies  this  substance 
is  distributed  continuously  and  abundantly  ;  and,  since  it  can 
c  33 


MEMOIRS    ON 

transmit  light,  we  are  led  to  expect  that  these  bodies  should  be 
as  transparent  as  glass,  which,  however,  is  not  the  fact. 

How,  then,  shall  we  explain  their  opacity?  Are  their  con- 
stituent particles  soft  and  built  up  of  still  smaller  particles, 
and  thus  able  to  change  shape  when  they  are  struck  by  ether 
particles  ?  Do  they  thus  damp  out  the  motion  and  stop  the 
propagation  of  the  light-waves  ?  This  seems  hardly  possible  ; 
for  if  the  particles  of  a  metal  were  soft,  how  could  polished 
silver  and  mercury  reflect  light  so  well  ?  What  seems  to  me 
more  probable  is  that  metallic  bodies,  which  are  almost  the 
only  ones  that  are  really  opaque,  have  interspersed  among  their 
hard  particles  some  which  are  soft,  the  former  producing  reflec- 
tion, the  latter  destroying  transparency ;  while,  on  the  other 
hand,  transparent  bodies  are  made  up  of  only  hard  and  elastic 
particles,  which,  together  with  the  ether,  propagate  light- waves 
in  the  manner  already  indicated. 

We  pass  now  to  the  explanation  of  refraction,  assuming,  as 
above,  that  light-waves  pass  through  transparent  substances 
arid  in  them  undergo  diminution  of  speed. 

The  fundamental  phenomenon  in  refraction  is  the  follow- 
ing, viz.,  when  any  ray  of  light,  AB,  travelling  in  air,  strikes 
obliquely  upon  the  polished  surface  of  a  transparent  body,  PGy 
it  undergoes  a  sudden  change  of  direction  at  the  point  of  inci- 
dence, B ;  and  this  change  occurs  in  such  a  way  that  the  angle 
CBE,  which  the  ray  makes  with  the  normal  to  the  surface,  is 
less  than  the  angle  ABD,  which  the  ray  in  air  made  with 
the  same  normal.  To  determine  the  numerical  value  of  these 
angles,  describe  about  the  point  B  a  circle  cutting  the  rays  AB, 
BC.  Then  the  perpendiculars,  AD,  CE, 
let  fall  from  these  points  of  intersection 
upon  the  normal,  DE,  viz.,  the  sines  of 
the  angles  ABD,  CBE,  bear  to  one  an- 
other a  certain  ratio  which,  for  any 
one  transparent  body,  is  constant  for 
all  directions  of  the  incident  ray.  For 
glass  this  ratio  is  almost  exactly  f, 
while  for  water  it  is  very  nearly  f ,  thus 
varying  from  one  transparent  body  to 
another. 

Another  property,  not  unlike  the  preceding,  is  that  the  refrac- 
tions of  rays  entering  and  of  rays  emerging  from  a  transpar- 

34 


THE    WAVE-THEORY    OF    LIGHT 


M 


Fig.  10 


ent  body  are  reciprocal.  That  is  to  say,  if  an  incident  ray,  AB, 
be  refracted  by  a  transparent  body  into  the  ray  BO,  so  also  will 
a  ray,  CB,  in  the  interior  of  the  body  be  refracted,  on  emer- 
gence, into  the  ray  BA. 

In  order  to  explain  these  phenomena  on  our  theory,  let  the 
straight  line  AB  Fig.  10,  represent  the  plane  surface  bounding  a 
transparentbodyextendingin 
a  direction  between  C  and  N. 

By  the  use  of  the  word 
plane  we  do  not  mean  to 
imply  a  perfectly  smooth  sur- 
face, but  merely  such  a  one 
as  was  employed  in  treating 
of  reflection,  and  for  the 
same  reason.  Let  the  line 
AC  represent  a  part  of  a 
light- wave  whose  source  is 
so  distant  that  this  part  may 
be  treated  as  a  straight  line. 
The  region  0,  on  the  wave 
AC,  will,  after  a  certain  in- 
terval of  time,  arrive  at  the  plane  AB,  along  the  straight  line 
CB,  which  we  must  think  of  #s  drawn /from  the  source  of 
light,  and  which  will,  therefore,  intersect  AC  at  right  angles. 
But  during  this  same  interval  of  time  the  region  about  A  would 
have  arrived  at  G,  along  the  straight  line  AG,  equal  and  parallel 
to  CB,  and,  indeed,  the  whole  of  the  wave  AC  would  have 
reached  the  position  GB,  provided  the  transparent  body  were 
capable  of  transmitting  waves  as  rapidly  as  the  ether.  But 
suppose  that  the  rate  of  transmission  is  less  rapid,  say  one-third 
less.  Then  the  motion  from  the  point  A  will  extend  into  the 
transparent  body  to  a  distance  which  is  only  two-thirds  of  CB, 
while  producing  its  secondary  spherical  wave  as  described 
above.  This  wave  is  represented  by  the  circle  SNR,  whose  cen- 
tre is  at  A  and  whose  radius  is  equal  to  f  CB.  If  we  consider, 
in  like  manner,  the  other  points  H  of  the  wave  AC,  it  will  be 
seen  that,  during  the  same  time  which  C  employs  in  going  to 
B,  these  points  will  not  only  have  reached  the  surface  AB,  along 
the  straight  lines  HK,  parallel  to  CB,  but  they  will  have  start- 
ed secondary  waves  into  the  transparent  body  from  the  points 
K  as  centres.  These  secondary  waves  are  represented  by  cir- 

35 


MEMOIRS    ON 


cles  whose  radii  are  respectively  equal  to  f  of  the  distances 
KM — that  is,  f  of  the  prolongations  of  HK  to  the  straight  line 
BG.  If  the  two  transparent  media  had  the  same  ability  to 
transmit  light,  these  radii  would  equal  the  whole  lengths  of 
the  various  lines  KM. 

But  all  these  circles  have  a  common  tangent  in  the  line  BN, 
viz.,  the  same  line  which  we  drew  from  the  point  B  tangent  to 
the  circle  SNR  first  considered.  For  it  is  easy  to  see  that  all 
the  other  circles  from  B  up  to  the  point  of  contact,  N,  touch, 
in  the  same  manner,  the  line  BN,  where  N  is  also  the  foot  of 
the  perpendicular  let  fall  from  A  upon  BN. 

We  may,  therefore,  say  that  BN  is  made  up  of  small  arcs  of 
these  circles,  and  that  it  marks  the  limits  which  the  motion 
from  the  wave  AC  has  reached  in  the  transparent  medium,  and 
the  region  where  this  motion  is  much  greater  than  anywhere 
else.  And,  furthermore,  that  this  line,  as  already  indicated,  is 
the  position  assumed  by  the  wave  AC  at  the  instant  when  the 
region  C  has  reached  the  point  B.  For  there  is  no  other  line 
below  the  plane  AB,  which,  like  BN,  is  a  common  tangent  to 
all  these  secondary  waves. 

Accordingly,  if  one  wishes  to  discover  through  what  in- 
termediate steps  the  wave  AC  reached  the  position  BN,  he  has 
only  to  draw,  in  the  same  figure,  the  straight  lines  KO 
parallel  to  BN,  and  all  the  lines  KL  parallel  to  ,AC.  He  will 
thus  see  that  the  wave  CA  passes  from  a  straight  line  into  the 
successive  broken  lines  LKO,  reassuming  the  form  of  a  straight 
line  in  the  position  BN.  From  what  has  preceded  this  will  be 
so  evident  as  to  need  no  further  explanation. 

If,   now,  using   the  same 

C  figure,  we  draw  EAF   nor- 

mal to  the  plane  AB  at  the 
point  A,  and  draw  DA  at 
right  angles  to  the  wave  AC, 
the  incident  ray  of  light  will 
then  be  represented  by  DA; 
and  AN,  which  is  drawn  per- 
pendicular to  BN,  will  be 
the  refracted  ray;  for  these 
rays  are  merely  the  straight 

lines  along  which  the  parts 

ffiff-  10  of  the  waves  travel. 

36 


THE    WAVE-THEORY    OF    LIGHT 


From  the  foregoing  it  is  easy  to  deduce  the  principal  law  of 
refraction,  viz.,  that  the  sine  of  the  angle  DAE  always  bears  a 
constant  ratio  of  the  sine  of  the  angle  NAF,  whatever  may  be 
the  direction  of  the  incident  ray,  and  that  the  ratio  is  the 
same  as  that  which  the  speed  of  the  waves  in  the  medium  on 
the  side  AE  bears  to  their  speed  on  the  side  AF. 

For  if  we  consider  AB  as  the  radius  of  a  circle,  the  sine 
of  the  angle  BAG  is  BC,  and  the  sine  of  the  angle  ABN  _is 
AN.  But  the  angles  BAG  and  DAE  are  equal;  for  each  is 
the  complement  of  CAE.  And  the  angle  ABN  is  equal  to 
NAF,  since  each  is  the  complement  of  BAN".  Hence  the  sine 
of  the  angle  DAE  is  to  the  sine  NAF  as  BC  is  to  AN.  But 
the  ratio  of  BO  to  AN  is  the  same  as  that  of  the  speeds  of 
light  in  the  media  on  the  side  towards  AE  and  the  side  tow- 
ards AF,  respectively ;  hence,  also,  the  sine  of  the  angle  DAE 
bears  to  the  sine  of  the  angle  NAF  the  same  ratio  as  these  two 
speeds  of  light. 

In  order  to  follow  the  refracted  ray  when  the  light-waves  en- 
ter a  body  which  transmits  them  .more  rapidly  than  the  body 
from  which  they  emerge  (say  in  the  ratio  of  3  to  2),  it  is  necessary 
only  to  repeat  the  same  construction  and  the  same  demonstra- 
tion which  we  have  just  been 
using,  substituting,  however, 
f  in  place  of  f .  And  we  find, 
by  the  same  logical  process, 
employing  this  other  figure, 
that  when  the  region  0  of 
the  wave  AO  reaches  the 
point  B  of  the  surface  AB, 
the  whole  wave  AC  will  have 
advanced  to  the  position  BN, 
such  that  the  ratio  of  BC,  per- 
pendicular to  AC,  is  to  AN, 
perpendicular  to  BN,  as  2  is 
to  3.  The  same  ratio  will 
also  hold  between  the  sine  of  the  angle  EAD  and  the  sine  of 
the  angle  FAN. 

The  reciprocal  relations  between  the  refractions  of  a  ray 
on  entering  and  on  emerging  from  one  and  the  same  me- 
dium is  thus  made  evident.  If  the  ray  NA  is  incident  upon 
the  exterior  surface  AB,  and  is  refracted  into  AD,  then 

37 


Fig.  11 


MEMOIRS    ON 

the  ray  DA  on  emerging  from  the  medium  will  be  refracted 
into  AN". 

We  are  now  able  to  explain  a  remarkable  phenomenon  which 
occurs  in  this  refraction.  When  the  incident  ray  DA  exceeds 
a  certain  inclination  it  loses  its  ability  to  pass  into  the  other 
medium.  Because  if  the  angle  DAQ  or  OBA  is  such  that,  in 
the  triangle  AOB,  OB  is  equal  to  or  greater  than  f  of  AB,  then 
AN,  being  equal  to  or  greater  than  AB,  can  no  longer  form  one 
side  of  the  triangle  ANB.  '  Therefore  the  wave  BN  does  not 
exist,  and  consequently  there  can  be  no  line  AN  drawn  at  right 
angles  to  it.  And  thus  the  incident  ray  DA  cannot  penetrate 
the  surface  AB. 

When  the  wave-speeds  are  in  the  ratio  of  %  to  3,  as  in  the 
case  of  glass  and  air,  which  we  have  considered,  the  angle 
DAQ  must  exceed  48°  11'  if  the  ray  DA  is  to  emerge.  And 
when  the  ratio  of  speeds  is  that  of  3  to  4,  as  is  almost  exactly 
the  case  in  water  and  air,  this  angle  DAQ  must  be  greater  than 
41°  24'.  And  this  agrees  perfectly  with  experiment. 

But  one  may  here  ask  why  no  light  penetrates  the  surface, 
since  the  encounter  of  the  wave  AC  against  the  surface  AB 
must  produce  some  motion  in  the  medium  on  the  other  side. 
The  answer  is  simple,  if  we  recall  what  has  already  been  said. 
For  although  an  infinite  number  of  secondary  waves  may  be 
started  into  the  medium  on  the  other  side  of  AB,  these  waves 
at  no  time  have  a  common  tangent*  line,  either  straight  or 
curved.  There  is  thus  no  line  which  marks  the  limit  to  which 
the  wave  AC  has  been  transmitted  beyond  the  plane, AB,  nor  is 
there  any  line  in  which  the  motion  has  been  sufficiently  con- 
centrated to  produce  light. 

In  the  following  manner  one  may  easily  recognize  the  fact 
that,  when  OB  is  greater  than  f  AB,  the  waves  beyond  the 
plane  AB  have  no  common  tangent.  About  the  centres  K 
describe  circles  having  radii  respectively  equal  to  f  LB.  These 
circles  will  enclose  one  another  and  will  each  pass  beyond  the 
point  B. 

It  is  to  be  noted  that  just  as  soon  as  the  angle  DAQ  becomes 
too  small  to  allow  the  refracted  ray  DA  to  pass  into,  the  other 
medium,  the  internal  reflection  which  occurs  at  the  surface  AB 
increases  rapidly  in  brilliancy,  as  may  be  easily  shown  by  means 
of  a  triangular  prism.  In  terms  of  our  theory,  we  may  thus 
explain  this  phenomenon :  While  the  angle  DAQ  is  still  large 

38 


THE    WAVE-THEORY    OF    LIGHT 

enough  for  the  ray  DA  to  be  transmitted,  it  is  evident  that  the 
light  from  the  wave-front  AC  will  be  concentrated  into  a  much 
shorter  line  when  it  reaches  the  position  BN.  It  will  be  seen 
also  that  the  wave-front  BN  grows  shorter  in  proportion  as  the 
angle  CBA  or  DAQ  becomes  smaller,  until  finally,  when  the 
limit  indicated  above  is  reached,  BN  is  reduced  to  a  point. 
That  is  to  say,  when  the  region  about  C,  on  the  wave  AC, 
reaches  B,  the  wave  BN",  which  is  the  wave  AC  after  trans- 
mission, is  entirely  compressed  into  this  same  point  B  ;  and,  in 
like  manner,  when  the  region  about  H  has  reached  the  point  K 
the  part  AH  is  completely  reduced  to  this  same  point  K.  It 
follows,  therefore,  that  in  proportion  as  the  direction  of  propa- 
gation of  the  wave  AC  happens  to  coincide  with  the  surface  AB, 
so  will  be  the  quantity  of  motion  along  this  surface. 

Now  this  motion  must  necessarily  spread  into  the  transparent 
body  and  also  greatly  reinforce  the  secondary  waves  which  pro- 
duce internal  reflection  at  the  face  AB,  according  to  the  laws 
of  this  reflection  explained  above. 

And  since  a  small  diminution  in  the  angle  of  incidence  is 
sufficient  to  reduce  the  wave -front  BN  from  a  fairly  large 
quantity  to  zero  (for  if  this  angle  in  the  case  of  glass  be 
49°  11',  the  angle  BAN  amounts  to  as  much  as  11°  21';  but  if 
this  same  angle  DAQ  be  diminished  by  one  degree  only,  the 
angle  BAN  becomes  zero  and  the  wave-front  BN  is  reduced  to 
a  point),  it  follows  that  the  internal  reflection  occurs  suddenly, 
passing  from  comparative  darkness  to  brilliancy  at  the  instant 
when  the  angle  of  incidence  assumes  a  value  which  no  longer 
permits  refraction. 

Now  as  to  ordinary  external  reflection,  i.  e.,  reflection  which 
occurs  when  the  angle  of  incidence  DAQ  is  still  large  enough 
to  allow  the  refracted  ray  to  pass  through  the  face  AB,  this 
reflection  must  be  from  the  particles  which  bound  the  trans- 
parent body  on  the  outside,  apparently  from  particles  of  air 
and  from  others  which  are  mixed  with,  but  are  larger  than,  the 
ether  particles. 

On  the  other  hand,  external  reflection  from  bodies  is  pro- 
duced by  the  particles  which  compose  these  bodies,  and  which 
are  larger  than  those  of  the  ether,  since  the  ether  flows  through 
the  interstices  of  the  body. 

It  must  be  confessed  that  we  here  find  difficulty  in  explain- 
ing the  experimental  fact  that  internal  reflection  occurs  even 

39 


MEMOIRS    ON 


where  the  particles  of  air  can  cut  no  figure,  as,  for  instance,  in 
vessels  and  tubes  from  which  the  air  has  been  exhausted. 

Experiment  shows  further  that  these  two  reflections  are  of 
almost  equal  intensity,  and  that  in  various  transparent  bodies 
this  intensity  increases  directly  as  the  refractive  index.  Thus 
we  see  that  reflection  from  glass  is  stronger  than  that  from 
water,  while  in  turn  that  from  diamond  is  stronger  than  that 
from  glass. 

I  shall  conclude  this  theory  of  refraction  by  demonstrating  a 
remarkable  proposition  depending  upon  it,  namely,  that  when 
a  ray  of  light  passes  from  one  point  to  another,  the  two  points 
lying  in  different  media,  refraction  at  the  bounding  surface 
occurs  in  such  a  way  as  to  make  the  time  required  the  least 
possible  ;  and  exactly  the  same  thing  occurs  in  reflection  at  a 
plane  surface.  M.  Fermat  discovered  this  property  of  refrac- 
tion, believing  with  us  and  in  opposition  to  M.  Descartes  that 
light  travels  more  slowly  through  glass  than  through  air. 
But,  besides  this,  he  assumed  what  we  have  just  proved  from 
the  fact  that  the  velocities  in  the  two  media  are  different,  viz., 
that  the  ratio  of  the  sines  is  a  constant ;  or,  what  amounts  to 
the  same  thing,  he  assumed,  besides  the  different  velocities, 
that  the  time  employed  was  a  minimum ;  and  from  this  he 
derived  the  constancy  of  the  sine  ratio. 

His*  demonstration,  which  may  be  found  in  his  works  and  in 
the  correspondence  of  M.  Descartes,  is  very  long.  It  is  for  this 
reason  that  I  here  offer  a  simpler  and  easier  one. 

Let  KF  represent  a  plane  surface  ;  imagine  the  point  A  in 
the  medium  through  which  the  light  travels  more  rapidly,  say 

air ;  the  point  C  lies  in  anoth- 
er, say  water,  in  which  the  speed 
of  light  is  less.  Let  us  sup- 
pose that  a  ray  passes  from  A, 
through  B,  to  0,  suffering  re- 
fraction at  B,  according  to  the 
law  above  demonstrated  ;  or, 
what  is  the  same  thing,  having 
drawn  PBQ  perpendicular  to 
the  surface,  the  sine  of  the 
angle  ABP  is  to  the  sine  of  the 
angle  CBQ  in  the  same  ratio  as 
the  speed  of  light  in  the  medium 
40 


THE    WAVE-THEORY    OF    LIGHT 

containing  A  is  to  the  speed  in  the  medium  containing  C.  It 
remains  to  show  that  the  time  required  for  the  light  to  traverse 
AB  and  BC  taken  together  is  the  least  possible.  Let  us  assume 
that  the  light  takes  some  other  path,  say  AF,  FC,  where  F,  the 
point  at  which  refraction  occurs,  is  more  distant  than  B  from  A. 
Draw  AO  perpendicular  to  AB,  and  FO  parallel  to  BA ;  BH 
perpendicular  to  FO,  and  FG  perpendicular  to  BC.  Since,  now, 
the  angle  HBF  is  equal  to  PBA,  and  the  angle  BFG  is  equal  to 
QBC,  it  follows  that  the  sine  of  the  angle  HBF  will  bear  to  the 
sine  of  BFG  the  same  ratio  as  the  speed  of  light  in  the  medium 
A  bears  to  the  speed  in  the  medium  0.  But  if  we  consider  BF 
the  radius  of  a  circle,  then  sines  are  represented  by  the  lines 
HF,  BG.  Accordingly,  the  lines  HF,  BG  are  in  the  ratio  of 
these  speeds.  If,  therefore,  we  imagine  OF  to  be  the  incident 
ray,  the  time  of  passage  from  H  to  F  will  be  the  same  as  the 
time  of  passage  from  B  to  G  in  the  medium  C. 

But  the  time  from  A  to  B  is  equal  to  the  time  from  0  to  H. 
Hence  the  time  from  0  to  F  is  the  same  as  the  time  from  A  to 
G,  via  B.  Again,  the  time  along  FC  is  greater  than  the  time 
along  GC  ;  and  hence  the  time  along  the  route  OFC  is  greater 
than  that  along  the  path  ABC.  But  AF  is  greater  than  OF; 
hence,  a  fortiorij  the  time  along  AFC  is  greater  than  that 
along  ABC. 

Let  us  now  assume  that  the  ray  passes  from  A  to  C  by  the 
route  AK,  KC,  the  point  of  refraction,  K,  being  nearer  to  A 
than  is  B.  Draw  CN  perpendicular  to  BC ;  KN"  parallel  to  BC ; 
BM  perpendicular  to  KN ;  and  KL  perpendicular  to  BA. 

Here  BL  and  KM  represent  the  sines  of  the  angles  BKL  and 
KBM — that  is,  the  angles  PBA  and  QBC  ;  and  hence  they  are 
in  the  same  ratio  as  the  speeds  of  light  in  the  media  A  and  C 
respectively.  The  time,  therefore,  from  L  to  B  is  equal  to  the 
time  from  K  to  M  ;  and,  since  the  time  from  B  to  C  is  equal  to 
the  time  from  M  to  N,  the  time  by  the  path  LBC  is  the  same 
as  the  time  via  KMN.  But  the  time  from  A  to  K  is  greater 
than  the  time  from  A  to  L,  and,  therefore,  the  time  by  the 
route  AKN  is  greater  than  the  route  ABC. 

Not  only  so,  but  since  KC  is  greater  than  KN,  the  time  by 
the  path  AKC  will  be  so  much  the  greater  than  by  the  path 
ABC.  Hence  follows  that  which  was  to  be  proved,  namely, 
that  the  time  along  the  path  ABC  is  the  least  possible. 

41 


MEMOIRS    ON 


BIOGRAPHICAL  SKETCH 

WHILE  there  are  no  sharp  lines  in  nature,  there  is  a  very 
true  sense  in  which  the  year  1642,  marking  the  death  of 
Galileo  and  the  birth  of  Newton,  serves  as  a  line  of  demar- 
cation between  the  foundation  and  the  superstructure  of  mod- 
ern physics. 

Galileo,  by  his  careful  study  of  gravitation,  by  his  clear  grasp 
of  force  as  determining  acceleration,  by  his  careful  search  after 
causes  and  their  respective  effects,  by  his  profound  faith  in 
experiment,  had  more  than  cleared  the  ground  for  the  build- 
ers of  modern  physics.  The  rapid  rise  of  this  structure  at 
the  hands  of  Newton  and  his  brilliant  contemporaries,  Boyle, 
Leibnitz,  Bonier,  Du  Fay,  Bradley,  and  Hooke,  marks  a  dis- 
tinctly modern  era  compared  with  that  of  Galileo. 

The  work  of  Christiaan  Huygens,  the  "Dutch  Archimedes," 
occupies,  as  regards  both  time  and  character,  a  position  inter- 
mediate between  these  two  periods.  He  was  born  at  The 
Hague  in  1629,  and  died  there  in  1695.  A  splendid  ancestry, 
three  years  of  university  training  at  Leyden  and  Breda,  much 
travel,  and  a  rare  group  of  associates,  combined  to  give  him 
an  education  which  left  little  to  be  desired.  Most  of  his  life 
was  spent  in  Holland,  but  for  the  fifteen  years  following  1666 
he  lived  and  worked  in  Paris,  where  he  was  the  guest  of  Louis 
XIV.  and  the  then  recently  founded  French  Academy  of  Sci- 
ences. This  was  for  him  a  happy  period  of  great  activity,  and 
it  was  only  in  anticipation  of  the  revocation  of  the  Edict  of 
Nantes,  in  1685,  that  he  returned  to  his  own  country,  where 
in  private  retirement  and  study  he  spent  most  of  his  remain- 
ing years. 

His  intellectual  achievements  fall  into  three  not  very  dis- 
tinct departments  of  science — namely,  mathematics,  physics, 
and  physical  astronomy.  In  mathematics,  his  chief  accom- 
plishments refer  to — 

(a)  The  quadrature  of  conies. 

(#)  The  theory  of  probabilities. 

(c)  A  discussion  of  the  evolutes  and  involutes  of  curves  and 
the  introduction  of  the  idea  of  the  envelope  of  a  moving 
straight  line. 

42 


THE    WAVE-THEORY     OF    LIGHT 

In  physics  he  gave — 

(a)  A  general  solution  of  the  problem  of  the  Compound  Pen- 

dulum, and  in  the  demonstration  enunciated  the  very 
general  principle  that  in  any  mechanical  system  acting 
under  gravity  the  centre  of  gravity  can  never  rise  to  a 
point  higher  than  that  from  which  it  fell — a  principle 
which  we  now  recognize  as  a  special  case  of  the  law 
that  the  potential  energy  of  any  mechanical  system  tends 
to  a  minimum. 

(b)  The  invention  of  the  pendulum  clock  and  its  application 

to  the  measurement  of  gravity  at  various  points  on  the 
earth's  surface. 

(c)  An  accurate   description  of  the   behavior  of  bodies  in 
collision. 

(d)  The  laws  governing  centrifugal  forces. 

(e)  The  undulatory  theory  of  light  arid  its  application  to 
the  explanation  of  reflection,  ordinary  refraction,  and 
double  refraction. 

Among  his  contributions  to  physical  astronomy  are — 

(a)  The  construction  of  the  first  powerful  telescope  of  the 
refracting  kind. 

(b)  The   discovery  of   the   rings   of   Saturn    and   its    sixth 

satellite. 

(c)  Improvements  in  the  methods  of  grinding  lenses  and  the 
addition  of  a  tube  to  the  object-glass  and  another  to 
the  eye-piece  of  the  aerial  telescope. 

All  his  mechanical  inventions  are  characterized  by  practica- 
bility, and  all  his  intellectual  work  by  clearness  and  elegance. 

Those  who  wish  a  more  detailed  account  of  his  activity  will 
find  it  in  the  superb  edition  of  his  works*  recently  published 
by  the  Societe  Hollandaise  des  Sciences,  while  that  delightful 
sketch  of  his  life  and  work  given 'by  Dr.  Bosschaf  should  be 
read  by  every  one. 

*  (Euvres  Computes  de  Christiaan  Huygens  (La  Haye  :  Martin  us  Nijhoff, 
1888  to  19—). 

f  Bosscha  :  Christiaan  Huygens,  Rede  am  200sten  Ged^chtnistage  seines 
Lebensende.  Ubersetzt  von  Engelmann.  (Eugelmann  :  Leipzig,  1895), 
pp.  77. 

43 


>*-'    ov 


LIBRA 


Oc 


ON  THE  THEORY  OF  LIGHT  AND 
COLOES 

From  the  Philosophical  Transactions  for  1802,  p.  12. 


AN  ACCOUNT  OF  SOME  CASES  OF  THE 

PRODUCTION   OF  COLORS  NOT 

HITHERTO  DESCRIBED 

From  the  Philosophical  Transactions  for  1802.  p.  387. 


EXPERIMENTS  AND  CALCULATIONS 
RELATIVE  TO  PHYSICAL  OPTICS 

From  the  Philosophical  Transactions  for  1804. 

.BY 

THOMAS   YOUNG. 


These  three  papers  are  reprinted  in  Young's  Miscellaneous  Works,  vol.  i., 
and  also  in  his  Lectures  on  Natural  Philosophy  and  Mechanical  Arts. 

45 


CONTENTS 

PAGE 

General  Statement  of  Wave  -  Theory,  including  the  Principle  of  Inter- 
ference       47 

Diffraction  in  che  Shadow  of  a  Narrow  Obstacle , 62 

Observations  on  the  Interference  Bands  in  the  Shadow  of  a  Narrow 
Obstacle. . .  68 


46 


ON  THE  THEORY  OF  LIGHT  AND 
COLORS* 

A   BAKERIAN   LECTURE 

Read  November  12,  1801. 


ALTHOUGH  the  invention  of  plausible  hypotheses,  indepen- 
dent of  any  connection  with  experimental  observations,  can  be  of 
very  little  use  in  the  promotion  of  natural  knowledge,  yet  the 
discovery  of  simple  and  uniform  principles,  by  .which  a  great 
number  of  apparently  heterogeneous  phenomena  are  reduced 
to  coherent  and  universal  laws,  must  ever  be  allowed  to  be  of 
considerable  importance  towards  the  improvement  of  the  hu- 
man intellect. 

The  object  of  the  present  dissertation  is  not  so  much  to  pro- 
pose any  opinions  which  are  absolutely  new,  as  to  refer  some 
theories,  which  have  been  already  advanced,  to  their  original 
inventors,  to  support  them  by  additional  evidence,  and  to  apply 
them  to  a  great  number  of  diversified  facts,  which  have  hither- 
to been  buried  in  obscurity.  Nor  is  it  absolutely  necessary  in 
this  instance  to  produce  a  single  new  experiment ;  for  of  ex- 
periments there  is  already  an  ample  store,  which  are  so  much 
the  more  unexceptionable  as  they  must  have  been  conducted 
without  the  least  partiality  for  the  system  by  which  they  will 
be  explained  ;  yet  some  facts,  hitherto  unobserved,  will  be 
brought  forward,  in  order  to  show  the  perfect  agreement  of 
that  system  with  the  multifarious  phenomena  of  nature. 

The  optical  observations  of  Newton  are  yet  unrivalled  ;  and, 
excepting  some  casual  inaccuracies,  they  only  rise  in  our  esti- 
mation as  we  compare  them  with  later  attempts  to  improve  on 

*  From  the  Philosophical  Transactions  for  1802,  p.  12 

47 


MEMOIRS    ON 

them.  A  further  consideration  of  the  colors  of  thin  plates,  as 
they  are  described  in  the  second  book  of  Newton's  Optics,  has 
converted  that  prepossession  which  I  before  entertained  for  the 
undulatory  system  of  light  into  a  very  strong  conviction  of  its 
truth  and  sufficiency,  a  conviction  which  has  been  since  most 
strikingly  confirmed  by  an  analysis  of  the  colors  of  striated 
substances.  The  phenomena  of  thin  plates  are  indeed  so  sin- 
gular that  their  general  complexion  is  not  without  great  diffi- 
culty reconcilable  to  any  theory,  however  complicated,  that 
has  hitherto  been  applied  to  them  ;  and  some  of  the  principal 
circumstances  have  never  been  explained  by  the  most  gratui- 
tous assumptions;  but  it  will  appear  that  the  minutest  particu- 
lars of  these  phenomena  are  not  only  perfectly  consistent  with 
the  theory  which  will  now  be  detailed,  but  that  they  are  all  the 
necessary  consequences  of  that  theory,  without  any  auxiliary 
suppositions ;  and  this  by  inferences  so  simple  that  they  be- 
come particular  corollaries,  which  scarcely  require  a  distinct 
enumeration. 

A  more  extensive  examination  of  Newton's  various  writings 
has  shown  me  that  he  was  in  reality  the  first  that  suggested 
such  a  theory  as  I  shall  endeavor  to  maintain  ;  that  his  own 
opinions  varied  less  from  this  theory  than  is  now  almost  univer- 
sally supposed  ;  and  that  a  variety  of  arguments  have  been  ad- 
vanced, as  if  to  confute  him,  which  may  be  found  nearly  in  a 
similar  form  in  his  own  works  ;  and  this  by  no  less  a  math- 
ematician than  Leonard  Euler,  whose  system  of  light,  as 
far  as  it  is  worthy  of  notice,  either  was,  or  might  have  been, 
wholly  borrowed  from  Newton,  Hooke,  Hnygens,  and  Male- 
branche. 

Those  who  are  attached,  as  they  may  be  with  the  greatest 
justice,  to  every  doctrine  which  is  stamped  with  the  Newtonian 
approbation,  will  probably  be  disposed  to  bestow  on  these  con- 
siderations so  much  the  more  of  their  attention,  as  they  appear 
to  coincide  more  nearly  with  Newton's  own  opinions.  For 
this  reason,  after  having  briefly  stated  each  particular  po- 
sition of  my  theory,  I  shall  collect,  from  Newton's  various 
writings,  such  passages  as  seem  to  be  the  most  favorable  to  its 
admission;  and  although  I  shall  quote  some  papers  which  may 
be  thought  to  have  been  partly  retracted  at  the  publication  of 
the  Optics,  yet  I  shall  borrow  nothing  from  them  that  can  be 
supposed  to  militate  against  his  rnaturer  judgment. 

48 


THE    WAVE-THEORY    OF    LIGHT 


HYPOTHESIS  I 

A  luminiferous  ether  pervades  the  universe,  rare  and  elastic 
in  a  high  degree. 

PASSAGES   FROM   KEWTOtf 

"The  hypothesis  certainly  has  a  much  greater  affinity  with 
his  own/'  that  is,  Dr.  Hooke's,  "  hypothesis  than  he  seems 
to  be  aware  of;  the  vibrations  of  the  ether  being  as  useful  and 
necessary  in  this  as  in  his." — Phil.  Trans.,  vol.  vii.,  p.  5087. 
Abr.,  vol.  i.,  p.  145.  Nov.,  1672. 

"  To  proceed  to  the  hypothesis  :  first,  it  is  to  be  supposed 
therein  that  there  is  an  ethereal  medium,  much  of  the  same 
constitution  with  air,  but  far  rarer,  subtler,  and  more  strongly 
elastic.  It  is  not  to  be  supposed  that  this  medium  is  one  uni- 
form matter,  but  compounded,  partly  of  the  main  phlegmatic 
body  of  ether,  partly  of  other  various  ethereal  spirits,  much 
after  the  manner  that  air  is  compounded  of  the  phlegmatic 
body  of  air,  intermixed  with  various  vapors  and  exhalations  : 
for  the  electric  and  magnetic  effluvia  and  gravitating  princi- 
ple seem  to  argue  such  variety." — BIRCH,  Hist,  of  R.  8.,  vol. 
iii.,  p.  249,  Dec.,  1675. 

"  Is  not  the  heat  (of  the  warm  room)  conveyed  through 
the  vacuum  by  the  vibrations  of  a  much  subtler  medium  than 
air  ?  And  is  not  this  medium  the  same  with  that  medium  by 
which  light  is  refracted  and  reflected,  and  by  whose  vibrations 
light  communicates  heat  to  bodies,  and  is  put  into  fits  of  easy 
reflection  and  easy  transmission  ?  And  do  not  the  vibrations 
of  this  medium  in  hot  bodies  contribute  to  the  intenseness  and 
duration  of  their  heat  ?  And  do  not  hot  bodies  communicate 
their  heat  to  contiguous  cold  ones  by  the  vibrations  of  this  me- 
dium propagated  from  them  into  the  cold  ones  ?  And  is  not  this 
medium  exceedingly  more  rare  and  subtle  than  the  air,  and  ex- 
ceedingly more  elastic  and  active  ?  And  doth  it  not  readily  per- 
vade all  bodies  ?  And  is  it  not,  by  its  elastic  force,  expanded 
through  all  the  heavens  ?  May  not  planets  and  comets,  and 
all  the  gross  bodies,  perform  their  motions  in  this  ethereal  me- 
dium ?  And  may  not  its  resistance  be  so  small  as  to  be  incon- 
siderable ?  For  instance,  if  this  ether  (for  so  I  will  call  it) 
should  be  supposed  700,000  times  more  elastic  than  our  air, 
and  above  700,000  times  more  rare,  its  resistance  would  be 
D  49 


MEMOIRS    ON 

about  600,000,000  times  less  than  that  of  water.  And  so  small 
a  resistance  would  scarce  make  any  sensible  alteration  in  the 
motions  of  the  planets  in  ten  thousand  years.  If  any  one 
would  ask  how  a  medium  can  be  so  rare,  let  him  tell  me  how 
an  electric  body  can  by  friction  emit  an  exhalation  so  rare  and 
subtle,  and  yet  so  potent?  And  how  the  effluvia  of  a  mag- 
net can  pass  through  a  plate  of  glass  without  resistance,  and 
yet  turn  a  magnetic  needle  beyond  the  glass  ?" — Optics, 
Qu.  18,  22. 

HYPOTHESIS  II 

Undulations  are  excited  in  this  ether  'whenever  a  body  becomes 
luminous. 

Scholium.  I  use  the  word  undulation  in  preference  to  vi- 
bration, because  vibration  is  generally  understood  as  implying 
a  motion  which  is  continued  alternately  backward  and  for- 
ward by  a  combination  of  the  momentum  of  the  body  with  an 
accelerating  force,  and  which  is  naturally  more  or  less  perma- 
nent; but  an  undulation  is  supposed  to  consist  in  vibratory 
motion  transmitted  successively  through  different  parts  of  a 
medium  without  any  tendency  in  each  particle  to  continue  its 
motion,  except  in  consequence  of  the  transmission  of  succeeding 
undulations  from  a  distinct  vibrating  body;  as  in  the  air  the 
vibrations  of  a  chord  produce  the  undulations  constituting 
sound. 

PASSAGES    FROM    NEWT02ST 

"  Were  I  to  assume  an  hypothesis,  it  should  be  this,  if  pro- 
pounded more  generally,  so  as  not  to  determine  what  light  is 
further  than  that  it  is  something  or  other  capable  of  e-xciting 
vibrations  in  the  ether  ;  for  thus  it  will  become  so  general  and 
comprehensive  of  other  hypotheses  as  to  leave  little  room  for 
new  ones  to  be  invented." — BIRCH,  Hist,  of  R.  S.,  vol.  iii.,  p. 
249.  Dec.,  1675. 

"  In  the  second  place,  it  is  to  be  supposed  that  the  ether  is  a 
vibrating  medium  like  air,  only  the  vibrations  far  more  swift 
and  minute  ;  those  of  air,  made  by  a  man's  ordinary  voice, 
succeeding  one  another  at  more  than  half  a  foot  or  a  foot  dis- 
tance, but  those  of  ether  at  a  less  distance  than  the  hundred- 
thousandth  part  of  an  inch.  And  as  in  air  the  vibrations  are 

50 


THE    WAVE-THEORY    OF    LI 

some  larger  than  others,  but  yet  all  equally  swift  (for  in  a  ring 
of  bells  the  sound  of  every  tone  is  heard  at  two  or  three  miles 
distance  in  the  same  order  that  the  bells  are  struck),  so,  I  sup- 
pose, the  ethereal  vibrations  differ  in  bigness,  but  not  in  swift- 
ness. Kow,  these  vibrations,  besides  their  use  in  reflection  and 
refraction,  may  be  supposed  the  chief  means  by  which  the  parts 
of  fermenting  or  putrefying  substances,  fluid  liquors,  or  melted, 
burning,  or  other  hot  bodies,  continue  in  motion." — BIRCH, 
Hist,  of  R.  S.,  vol.  iii.,  p.  251,  Dec.,  1675. 

"When  a  ray  of  light  falls  upon  the  surface  of  any  pellucid 
body,  and  is  there  refracted  or  reflected,  may  not  waves  of 
vibrations,  or  tremors,  be  thereby  excited  in  the  refracting  or 
reflecting  medium  ?  And  are  not  these  vibrations  propagated 
from  the  point  of  incidence  to  great  distances  ?  And  do  they 
not  overtake  the  rays  of  light,  and  by  overtaking  them  succes- 
sively, do  not  they  put  them  into  the  fits  of  easy  reflection  and 
easy  transmission  described  above  ?" — Optics,  Qu.  17. 

"Light  is  in  fits  of  easy  reflection  and  easy  transmission 
before  its  incidence  on  transparent  bodies.  And  probably  it  is 
put  into  such  fits  at  its  first  emission  from  luminous  bodies,  and 
continues  in  them  during  all  its  progress/'  —  Optics,  Book  ii., 
part  iii.,  prop.  13. 

HYPOTHESIS  III 

The  sensation  of  different  colors  depends  on  the  different 
frequency  of  vibrations  excited  by  light  in  the  retina. 

PASSAGES   FROM    NEWTON 

"The  objector's  hypothesis,  as  to  the  fundamental  part  of  it, 
is  not  against  me.  That  fundamental  supposition  is,  that  the 
parts  of  bodies,  when  briskly  agitated,  do  excite  vibrations  in 
the  ether,  which  are  propagated  every  way  from  those  bodies  in 
straight  lines,  and  cause  a  sensation  of  light  by  beating  and 
dashing  against  the  bottom  of  the  eye,  something  after  the 
manner  that  vibrations  in  the  air  cause  a  sensation  of  sound 
by  beating  against  the  organs  of  hearing.  Now,  the  most  free 
and  natural  application  of  this  hypothesis  to  the  solution  of 
phenomena  I  take  to  be  this — that  the  agitated  parts  of  bodies, 
according  to  their  several  sizes,  figures,  and  motions,  do  excite 
vibrations  in  the  ether  of  various  depths  or  bignesses,  which, 

51 


MEMOIRS    ON 

being  promiscuously  propagated  through  that  medium  to  our 
eyes,  effect  in  us  a  sensation  of  light  of  a  white  color;  but  if 
by  any  means  those  of  unequal  bignesses  be  separated  from  one 
another,  the  largest  beget  a  sensation  of  a  red  color;  the  least, 
or  shortest,  of  a  deep  violet,  and  the  intermediate  ones  of  inter- 
mediate colors ;  much  after  the  manner  that  bodies,  according 
to  their  several  sizes,  shapes,  and  motions,  excite  vibrations  in 
the  air  of  various  bignesses,  which,  according  to  those  bignesses, 
make  several  tones  in  sound  :  that  the  largest  vibrations  are 
best  able  to  overcome  the  resistance  of  a  refracting  superficies, 
and  so  break  through  it  with  least  refraction  ;  whence  the  vi- 
brations of  several  bignesses — that  is,  the  rays  of  several  colors, 
which  are  blended  together  in  light — must  be  parted  from  one 
another  by  refraction,  and  so  cause  the  phenomena  of  prisms 
and  other  refracting  substances  ;  and  that  it  depends  on  the 
thickness  of  a  thin  transparent  plate  or  bubble  whether  a  vi- 
bration shall  be  reflected  at  its  further  superficies  or  transmit- 
ted ;  so  that,  according  to  the  number  of  vibrations  interced- 
ing the  two  superficies,  they  may  be  reflected  or  transmitted  for 
many  successive  thicknesses.  And  since  the  vibrations  which 
make  blue  and  violet  are  supposed  shorter  than  those  which 
make  red  and  yellow,  they  must  be  reflected  at  a  less  thickness 
of  the  plate,  which  is  sufficient  to  explicate  all  the  ordinary 
phenomena  of  those  plates  or  bubbles,  and  also  of  all  natural 
bodies,  whose  parts  are  like  so  many  fragments  of  such  plates. 
These  seem  to  be  the  most  plain,  genuine,  and  necessary  con- 
ditions of  this  hypothesis;  and  they  agree  so  justly  with  my 
theory  that  if  the  animadversor  think  fit  to  apply  them,  he 
need  not,  on  that  account,  apprehend  a  divorce  from  it ;  but 
yet,  how  he  will  defend  it  from  other  difficulties  I  know  not."- 
Phil  Trans.,  vol.  vii.,  p.  5088.  Abr.,  vol.  i.,  p.  145.  Nov.,  1672. 
"To  explain  colors,  I  suppose  that  as  bodies  of  various 
sizes,  densities,  or  sensations  do  by  percussion  or  other  action 
excite  sounds  of  various  tones,  and  consequently  vibrations  in 
the  air  of  different  bigness,  so  the  rays  of  light,  by  impinging 
on  the  stiff  refracting  superficies,  excite  vibrations  in  the 
ether  of  various  bigness,  the  biggest,  strongest,  or  most  po- 
tent rays,  the  largest  vibrations  ;  and  others  shorter,  according 
to  their  bigness,  strength,  or  power  :  and  therefore  the  ends  of 
the  capillamenta  of  the  optic  nerve,  which  pave  or  face  the  ret- 
ina, being  such  refracting  superficies,  when  the  rays  impinge 

52 


THE    WAVE-THEORY    OF    LIGHT 

upon  them,  they  must  there  excite  these  vibrations,  which  vi- 
brations (like  those  of  sound  in  a  trunk  or  trumpet)  will  run 
along  the  aqueous  pores  or  crystalline  pith  of  the  capillamenta, 
through  the  optic  nerves,  into  the  sensorium  ;  and  there,  I 
suppose,  affect  the  sense  with  various  colors,  according  to  their 
bigness  and  mixture ;  the  biggest  with  the  strongest  colors, 
reds  and  yellows ;  the  least  with  the  weakest — blues  and  violets ; 
the  middle  with  green,  and  a  confusion  of  all  with  white — 
much  after  the  manner  that,  in  the  sense  of  hearing,  nature 
makes  use  of  aerial  vibrations  of  several  bignesses  to  generate 
sounds  of  divers  tones,  for  the  analogy  of  nature  is  to  be  ob- 
served."—BIRCH,  Hist,  of  R.  8.,  vol.  iii.,  p.  262,  Dec.,  1675. 

' '  Considering  the  lastingness  of  the  motions  excited  in  the  bot- 
tom of  the  eye  by  light,  are  they  not  of  a  vibrating  nature  ?  Do 
not  the  most  refrangible  rays  excite  the  shortest  vibrations,  the 
least  refrangible  the  largest  ?  May  not  the  harmony  and  dis- 
cord of  colors  arise  from  the  proportions  of  the  vibrations 
propagated  through  the  fibres  of  the  optic  nerve  into  the  brain, 
as  the  harmony  and  discord  of  sounds  arise  from  the  propor- 
tions of  the  vibrations  of  the  air  ?"—  Optics,  Qu.  16,  13,  14. 

[Scholium  omitted.] 


HYPOTHESIS  IV 

All  material  bodies  have  an  attraction  for  the  ethereal  medium, 
by  means  of  which  it  is  accumulated  within  their  substance, 
and  for  a  small  distance  around  them,  in  a  state  of  greater 
density  but  not  of  greater  elasticity. 

It  has  been  shown  that  the  three  former  hypotheses,  which 
may  be  called  essential,  are  literally  parts  of  the  more  compli- 
cated Newtonian  system.  This  fourth  hypothesis  differs  per- 
haps, in  some  degree  from  any  that  have  been  proposed  by 
former  authors,  and  is  diametrically  opposite  to  that  of  New- 
ton ;  but  both  being  in  themselves  equally  probable,  the  op- 
position is  merely  accidental,  and  it  is  only  to  be  inquired 
which  is  the  best  capable  of  explaining  the  phenomena.  Other 
suppositions  might  perhaps  be  substituted  for  this,  and  there- 
fore I  do  not  consider  it  as  fundamental,  yet  it  appears  to  be 
the  simplest  and  best  of  any  that  have  occurred  to  me. 

53 


OK    THli 

UNIVERSIT 


MEMOIRS    ON 


PROPOSITION   I 

All  impulses  are  propagated  in  a  homogeneous  elastic  medium 
with  an  equable  velocity. 

Every  experiment  relative  to  sound  coincides  with  the  ob- 
servation already  quoted  from  Newton,  that  all  undulations 
are  propagated  through  the  air  with  equal  velocity  ;  and  this  is 
further  confirmed  by  calculations.  (LAGRANGE,  Misc.  Taur., 
vol.  i.,  p.  91.  Also, -much  more  concisely,  in  my  syllabus  of 
a  course  of  lectures  on  Natural  and  Experimental  Philosophy, 
about  to  be  published.  Art.  289.)  If  the  impulse  be  so 
great  as  materially  to  disturb  the  density  of  the  medium,  it 
will  be  no  longer  homogeneous;  but,  as  far  as  concerns  our 
senses,  the  quantity  of  motion  may  be  considered  as  infinitely 
small.  It  is  surprising  that  Euler,  although  aware  of  the  mat- 
ter of  fact,  should  still  have  maintained  that  the  more  frequent 
undulations  are  more  rapidly  propagated.  (Tlieor.  mus.  and 
Conject.  phys.}  It  is  possible  that  the  actual  velocity  of  the 
particles  of  the  luminiferous  ether  may  bear  a  much  less  pro- 
portion to  the  velocity  of  the  undulations  than  in  sound,  for 
light  maybe  excited  by  the  motion  of  a  body  moving  at  the  rate 
of  only  one  mile  in  the  time  that  light  moves  a  hundred  millions. 

Scholium  1.  It  has  been  demonstrated  that  in  different  me- 
diums the  velocity  varies  in  the  snbduplicate  ratio  of  the  force 
directly  and  of  the  density  inversely.  (Misc.  Taur.,  vol.  L, 
p.  91.  Young's  Syllabus.  Art.  294.) 

Scholium  2.  It  is  obvious,  from  the  phenomena  of  elastic 
bodies  and  of  sounds,  that  the  undulations  may  cross  each 
other  without  interruption  ;  but  there  is  no  necessity  that  the 
various  colors  of  white  light  should  intermix  their  undula- 
tions, for,  supposing  the  vibrations  of  the  retina  to  continue 
but  a  five-hundredth  of  a  second  after  their  excitement,  a  mill- 
ion undulations  of  each  of  a  million  colors  may  arrive  in  dis- 
tinct succession  within  this  interval  of  time,  and  produce  the 
same  sensible  effect  as  if  all  the  colors  arrived  precisely  at  the 
same  instant. 

PROPOSITION  II 

An  undulation  conceived  to  originate  from  the  vibration  of  a 
single  particle  must  expand  through  a  homogeneous  medium 

54 


THE    WAVE-THEORY    OF    LIGHT 

in  a   spherical  form,  but  with  different  quantities  of  motion 
in  different  parts. 

For,  since  every  impulse,  considered  as  positive  or  negative, 
is  propagated  with  a  constant  velocity,  each  part  of  the  undula- 
tion must  in  equal  times  have  passed  through  equal  distances 
from  the  vibrating-point.  And,  supposing  the  vibrating  particle, 
in  the  course  of  its  motion,  to  proceed  forward  to  a  small  dis- 
tance in  a  given  direction,  the  principal  strength  of  the  undula- 
tion will  naturally  be  straight  before  it  ;  behind  it  the  motion 
will  be  equal  in  a  contrary  direction  ;  and  at  right  angles  to 
the  line  of  vibration  the  undulation  will  be  evanescent. 

Now,  in  order  that  such  an  undulation  may  continue  its 
progress  to  any  considerable  distance,  there  must  be  in  each 
part  of  it  a  tendency  to  preserve  its  own  motion  in  a  right  line 
from  the  centre ;  for  if  the  excess  of  force  at  any  part  were 
communicated  to  the  neighboring  particles,  there  can  be  no 
reason  why  it  should  not  very  soon  be  equalized  throughout, 
or,  in  other  words,  become  wholly  extinct,  since  the  motions  in 
contrary  directions  would  naturally  destroy  each  other.  The 
origin  of  sound  from  the  vibration  of  a  chord  is  evidently  of 
this  nature  ;  on  the  contrary,  in  a  circular  wave  of  water  every 
part  is  at  the  same  instant  either  elevated  or  depressed.  It  may 
be  difficult  to  show  mathematically  the  mode  in  which  this  in- 
equality of  force  is  preserved,  but  the  inference  from  the  mat- 
ter of  fact  appears  to  be  unavoidable ;  and  while  the  science  of 
hydrodynamics  is  so  imperfect  that  we  cannot  even  solve  the 
simple  problem  of  the  time  required  to  empty  a  vessel  by  ;i 
given  aperture,  it  cannot  be  expected  that  we  should  be  able 
to  account  perfectly  for  so  complicated  a  series  of  phenomena 
as  those  of  elastic  fluids.  The  theory  of  Huygens,  indeed,  ex- 
plains the  circumstance  in  a  manner  tolerably  satisfactory.  He 
supposes  every  particle  of  the  medium  to  propagate  a  distinct 
undulation  in  all  directions,  and  that  the  general  effect  is  only 
perceptible  where  a  portion  of  each  undulation  conspires  in 
direction  at  the  same  instant  ;  and  it  is  easy  to  show  that  sucli 
a  general  undulation  would  in  all  cases  proceed  rectilinearly, 
with  proportionate  force  ;  but,  upon  this  supposition,  it  seems 
to  follow  that  a  greater  quantity  of  force  must  be  lost  by  the 
divergence  of  the  partial  undulations  than  appears  to  be  con- 
sistent with  the  propagation  of  the  effect  to  any  considerable 

55 


MEMOIRS    ON 

distance ;  yet  it  is  obvious  that  some  such  limitation  of  the 
motion  must  naturally  be  expected  to  take  place,  for  if  the  in- 
tensity of  the  motion  of  any  particular  part,  instead  of  co^itinu- 
ing  to  be  propagated  straight  forward,  were  supposed  to  affect 
the  intensity  of  a  neighboring  part  of  the  undulation,  an  im- 
pulse must  then  have  travelled  from  an  internal  to  an  external 
circle  in  an  oblique  direction,  in  the  same  time  as  in  the  direc- 
tion of  the  radius,  and  consequently  with  a  greater  velocity, 
against  the  first  proposition.  In  the  case  of  water  the  velocity 
is  by  no  means  so  rigidly  limited  as  in  that  of  an  elastic  medium. 
Yet  it  is  not  necessary  to  suppose,  nor  is  it  indeed  probable,  that 
there  is  absolutely  not  the  least  lateral  communication  of  the 
force  of  the  undulation,  but  that,  in  highly  elastic  mediums, 
this  communication  is  almost  insensible.  In  the  air,  if  a  chord 
be  perfectly  insulated  so  as  to  propagate  exactly  such  vibra- 
tions as  have  been  described,  they  will,  in  fact,  be  much  less 
forcible  than  if  the  chord  be  placed  in  the  neighborhood  of  a 
sounding-board,  and  probably  in  some  measure  because  of  this 
lateral  communication  of  motions  of  an  opposite  tendency. 
And  the  different  intensity  of  different  parts  of  the  same  cir- 
cular undulation  may  be  observed  by  holding  a  common  tun- 
ing-fork at  arm's-length,  while  sounding,  and  turning  it,  from 
a  plane  directed  to  the  ear,  into  a  position  perpendicular  to 
that  plane. 

PROPOSITION  III 

A  portion  of  a  spherical  undulation,  admitted  through  an 
aperture  into  a  quiescent  medium,  toill  proceed  to  be  further 
propagated  rectilinearly  in  concentric  superficies,  terminated 
laterally  by  weak  and  irregular  portions  of  newly  diverging 
undulations. 

At  the  instant  of  admission  the  circumference  of  each  of 
the  undulations  may  be  supposed  to  generate  a  partial  undula- 
tion, filling  up  the  nascent  angle  between  the  radii  and  the  sur- 
face terminating  the  medium ;  but  no  sensible  addition  will  be 
made  to  its  strength  by  a  divergence  of  motion  from  any  other 
parts  of  the  undulation,  for  want  of  a  coincidence  in  time,  as 
has  already  been  explained  with  respect  to  the  various  force  of 
a  spherical  undulation.  If,  indeed,  the  aperture  bear  but  a  small 
proportion  to  the  breadth  of  an  undulation,  the  newly  gener- 

56 


THE    WAVE-THEORY    OF    LIGHT 


ated  undulation  may  nearly  absorb  the  whole  force  of  the  por- 
tion admitted ;  and  this  is  the  case  considered  by  Newton  in 
the  Principia.  But  no  experiment  can  be  made  under  these 
circumstanced  with  light,  on  account  of  the  minuteness  of  its 
undulations  and  the  interference  of  inflection;  and  yet  some 
faint  radiations  do  actually  diverge  beyond  any  probable  lim- 
its of  inflection,  rendering  the  margin  of  the  aperture  distinctly 
visible  in  all  directions.  These  are  attributed  by  Newton  to 
some  unknown  cause,  distinct  from  inflection  (Optics,  Book 
iii.,  obs.  5)/and  they  fully  answer  the  description  of  this 
proposition. 

Let  the  concentric  lines  in  Fig.  13  represent  the  con- 
temporaneous situation  of  similar  parts  of  a  number  of  succes- 
sive undulations  diverging  from  the  point  A;  they  will  also 
represent  the  successive  situations  of  each  individual  undula- 
tion: let  the  force  of  each  undulation  be  represented  by  the 
breadth  of  the  line,  and  let  the  cone  of  light  ABO  be  admitted 
through  the  aperture  BO;  then  the  principal  undulations  will 
proceed  in  a  rectilinear  direction  towards  GrH,  and  the  faint 
radiations  on  each  side  will  diverge  from  B 
and  0  as  centres,  without  receiving  any  ad- 
ditional force  from  any  intermediate  point 
D  of  the  undulation,  on  account  of  the  in- 
equality of  the  lines  DE  and  DF.  But  if 
we  allow  some  little  lateral  divergence  from 
the  extremities  of  the  undulations,  it  must 
diminish  their  force,  without  adding  materi- 
ally to  that  of  the  dissipated  light ;  and  their  - 
termination,  instead  of  the  right  line  BG, 
will  assume  the  form  OH,  since  the  loss  of 
force  must  be  more  considerable  near  to  0 
than  at  greater  distances.  This  line  corre- 
sponds with  the  boundary  of  the  shadow 
in  Newton's  first  observation,  Fig.  13  ;  and 
it  is  much  more  probable  that  such  a  dissi- 
pation of  light  was  the  cause  of  the  increase 
of  the  shadow  in  that  observation  than  that 
it  was  owing  to  the  action  of  the  inflecting 
atmosphere,  which  must  have  extended  a 
thirtieth  of  an  inch  each  way  in  order  to  pro- 
duce it ;  especially  when  it  is  considered  that 

57 


Fig.  13 


MEMOIRS    ON 

the  shadow  was  not  diminished  by  surrounding  the  hair  with 
a  denser  medium  than  air,  which  must  in  all  probability  have 
weakened  and  contracted  its  inflecting  atmosphere.  In  other 
circumstances  the  lateral  divergence  might  appear  to  increase, 
instead  of  diminishing,  the  breadth  of  the  beam. 

As  the  subject  of  this  proposition  has  always  been  esteemed 
the  most  difficult  part  of  the  undulatory  system,  it  will  be 
proper  to  examine  here  the  objections  which  Newton  has 
grounded  upon  it. 

''To  me  the  fundamental  supposition  itself  seems  impossi- 
ble— namely,  that  the  waves  or  vibrations  of  any  fluid  can,  like 
the  rays  of  light,  be  propagated  in  straight  lines,  without  a 
continual  and  very  extravagant  spreading  and  bending  every 
way  into  the  quiescent  medium,  where  they  are  terminated  by 
it.  I  mistake  if  there  be  not  both  experiment  and  demonstra- 
tion to  the  contrary." — Phil.  Trans.,  vol.  vii.,  p.  5089.  Abr., 
vol.  L,  p.  146.  Nov.  1672. 

"  Motus  omnis  per  flu  id  urn.  propagatus  divergit  a  recto 
tramite  in  spatia  immota.", 

"Quoniam  medium  ibi,"  in  the  middle  of  an  undulation  ad- 
mitted, "  densius  est,  quam  in  spatiis  hinc  inde,  dilatabit  sese 
tarn  versus  spatia  utrinque  sita,  quam  versus  pulsum  rariora 
intervalla ;  eoque  pacto — pulsns  eadem  fere  celeritate  sese  in 
medii  partes  quiescentes  hinc  inde  relaxare  debent ; — ideoqne 
spatium  totum  occupabunt — Hoc  experimur  in  sonis."— Prin- 
cip.,  lib.  ii.,  prop.  42. 

"Are  not  all  hypotheses  erroneous  in  which  light  is  sup- 
posed to  consist  in  pression  or  motion  propagated  through  a 
fluid  medium  ?  If  it  consisted  in  pression  or  motion,  propa- 
gated either  in  an  instant,  or  in  time,  it  would  bend  into  the 
shadow.  For  pression  or  motion  cannot  be  propagated  in  a 
fluid  in  right  lines  beyond  an  obstacle  which  stops  part  of  the 
motion,  but  will  bend  and  spread  every  way  into  the  quiescent 
medium  which  lies  beyond  the  obstacle.  The  waves  on  the 
surface  of  stagnating  water  passing  by  the  sides  of  a  broad  ob- 
stacle which  stops  part  of  them,  bend  afterwards,  and  dilate 
themselves  gradually  into  the  quiet  water  behind  the  obstacle. 
The  waves,  pulses,  or  vibrations  of  the  air,  wherein  sounds 
consist,  bend  manifestly,  though  not  so  much  as  the  waves  of 
water.  For  a  bell  or  a  cannon  may  be  heard  beyond  a  hill 
which  intercepts  the  sight  of  the  sounding  body  ;  and  sounds 

58 


THE    WAVE-THEORY    OF    LIGHT 

are  propagated  as  readily  through  crooked  pipes  as  straight 
ones.  Bat  light  is  never  known  to  follow  crooked  passages 
nor  to  bend  into  the  shadow.  For  the  fixed  stars,  by  the  inter- 
position of  any  of  the  planets,  cease  to  be  seen.  And  so  do 
the  parts  of  the  sun  by  the  interposition  of  the  moon,  Mer- 
cury, or  Venus.  The  rays  which  pass  very  near  to  the  edges 
of  any  body  are  bent  a  little  by  the  action  of  the  body  ;  but 
this  bending  is  not  towards  but  from  the  shadow,  and  is  per- 
formed only  in  the  passage  of  the  ray  by  the  body,  and  at  a 
very  small  distance  from  it.  So  soon  as  the  ray  is  past  the 
body  it  goes  right  on." — Optics,  Qu.  28. 

Now  the  proposition  quoted  from  the  Principia  does  not  di- 
rectly contradict  this  proposition  ;  for  it  does  not  assert  that 
such  a  motion  must  diverge  equally  in  all  directions  ;  neither 
can  it  with  truth  be  maintained  that  the  parts  of  an  elastic 
medium  communicating  any  motion  must  propagate  that  mo- 
tion equally  in  all  directions.  All  that  can  be  inferred  by  rea- 
soning is  that  the  marginal  parts  of  the  undulation  must  be 
somewhat  weakened  and  that  there  must  be  a  faint  divergence 
in  every  direction  ;  but  whether  either  of  these  effects  might 
be  of  sufficient  magnitude  to  be  sensible  could  not  have  been 
inferred  from  argument,  if  the  affirmative  had  not  been  ren- 
dered probable  by  experiment. 

As  to  the  analogy  with  other  fluids,  the  most  natural  infer- 
ence from  it  is  this  :  "The  waves  of  the  air,  wherein  sounds 
consist,  bend  manifestly,  though  not  so  much  as  the  waves  of 
water  ";  water  being  an  inelastic  and  air  a  moderately  elastic 
medium  ;  but  ether  being  most  highly  elastic,  its  waves  bend 
very  far  less  than  those  of  the  air,  and  therefore  almost  imper- 
ceptibly. Sounds  are  propagated  through  crooked  passages, 
because  their  sides  are  capable  of  reflecting  sound,  just  as  light 
would  be  propagated  through  a  bent  tube,  if  perfectly  polished 
within. 

The  light  of  a  star  is  by  far  too  weak  to  produce,  by  its  faint 
divergence,  any  visible  illumination  of  the  margin  of  a  planet 
eclipsing  it ;  and  the  interception  of  the  sun's  light  by  the 
moon  is  as  foreign  to  the  question  as  the  statement  of  inflec- 
tion is  inaccurate. 

To  the  argument  adduced  by  Huygens  in  favor  of  the  rec- 
tilinear propagation  of  undulations  Newton  has  made  no  reply; 
perhaps  because  of  his  own  misconception  of  the  nature  of  the 

59 


MEMOIRS    ON 

motions  of  elastic  mediums,  as  dependent  on  a  peculiar  law  of 
vibration,  which  has  been  corrected  by  later  mathematicians. 
On  the  whole,  it  is  presumed  that  this  proposition  may  be 
safely  admitted  as  perfectly  consistent  with  analogy  and  with 
experiment. 

PROPOSITION    IV 

When  an  undulation  arrives  at  a  surface  which  is  the  limit 
of  mediums  of  different  densities,  a  partial  reflection  takes 
place  proportionate  in  force  to  the  difference  of  the  densities. 

This  may  be  illustrated,  if  not  demonstrated,  by  the  analogy 
of  elastic  bodies  of  different  sizes.  "  If  a  smaller  elastic  body 
strikes  against  a  larger  one,  it  is  well  known  that  the  smaller  is 
reflected  more  or  less  powerfully,  according  to  the  difference  of 
their  magnitudes  :  thus,  there  is  always  a  reflection  when  the 
rays  of  light  pass  from  a  rarer  to  a  denser  stratum  of  ether  ; 
and  frequently  an  echo  when  a  sound  strikes  against  a  cloud. 
A  greater  body  striking  a  smaller  one  propels  it,  without  losing 
all  its  motion  :  thus,  the  particles  of  a  denser  stratum  of  ether 
do  not  impart  the  whole  of  their  motion  to  a  rarer,  but,  in  their 
effort  to  proceed,  they  are  recalled  by  the  attraction  of  the 
refracting  substance  with  equal  force  ;  and  thus  a  reflection  is 
always  secondarily  produced  when  the  rays  of  light  pass  from 
a  denser  to  a  rarer  stratum."  But  it  is  not  absolutely  necessary 
to  suppose  an  attraction  in  the  latter  case,  since  the  effort 
to  proceed  would  be  propagated  backward  without  it,  and  the 
undulation  would  be  reversed,  a  rarefaction  returning  in  place 
of  a  condensation  ;  and  this  will  perhaps  be  found  most  con- 
sistent with  the  phenomena. 

[Propositions  F.,  VI.,  and  VII.  omitted.] 

PROPOSITION  VIII 

When  two  undulations,  from  different  origins,  coincide 
either  perfectly  or  very  nearly  in  direction,  their  joint  effect 
is  a  combination  of  the  motions  belonging  to  each. 

Since  every  particle  of  the  medium  is  affected  by  each  undu- 
lation, wherever  the  directions  coincide,  the  undulations  can 
proceed  no  otherwise  than  by  uniting  their  motions,  so  that 
the  joint  motion  may  be  the  sum  or  difference  of  the  separate 

60 


THE    WAVE-THEORY    OF    LIGHT 

motions,  accordingly  as  similar  or  dissimilar  parts  of  the  undu- 
lations are  coincident. 

I  have,  on  a  former  occasion,,  insisted  at  large  on  the  appli- 
cation of  this  principle  to  harmonics  ;  and  it  will  appear  to  be 
of  still  more  extensive  utility  in  explaining  the  phenomena  of 
colors.  The  undulations  which  are  now  to  be  compared  are 
those  of  equal  frequency.  When  the  two  series  coincide  ex- 
actly in  point  of  time,  it  is  obvious  that  the  united  velocity  of 
the  particular  motions  must  be  greatest,  and,  in  effect  at  least, 
double  the  separate  velocities  ;  and  also  that  it  must  be  smallest, 
and,  if  the  undulations  are  of  equal  strength,  totally  destroyed 
when  the  time  of  the  greatest  direct  motion  belonging  to  one 
undulation  coincides  with  that  of  the  greatest  retrograde  motion 
of  the  other.  In  intermediate  states  the  joint  undulation  will 
be  of  intermediate  strength  ;  but  by  what  laws  this  intermediate 
strength  must  vary  cannot  be  determined  without  further  data. 
It  is  well  known  that  a  similar  cause  produces  in  sound  that 
effect  which  is  called  a  beat ;  two  series  of  undulations  of  nearly 
equal  magnitude  co-operating  and  destroying  each  other  alter- 
nately, as  they  coincide  more  or  less  perfectly  in  the  times  of 
performing  their  respective  motions. 

[Proposition  IX.  and  five  corollaries  to  Proposition  VIII. 
are  here  omitted.] 

61 


AN  ACCOUNT  OF  SOME  CASES  OF  THE 

PRODUCTION  OF  COLORS  NOT 

HITHERTO  DESCRIBED* 

READ  JULY  1,  1802 


WHATEVER  opinion  maybe  entertained  of  the  theory  o 
and  colors  which  I  have  lately  had  the  honor  of  submitting 
to  the  Royal  Society,  it  must  at  any  rate  be  allowed  that  it 
has  given  birth  to  the  discovery  of  a  simple  and  general  law 
capable  of  explaining  a  number  of  the  phenomena  of  col- 
ored light,  which,  without  this  law,  would  remain  insulated 
and  unintelligible.  The  law  is,  that  "  wherever  two  portions 
of  the  same  light  arrive  at  the  eye  by  different  routes,  either 
exactly  or  very  nearly  in  the  same  direction,  the  light  becomes 
most  intense  when  the  difference  of  the  routes  is  any  multiple 
of  a  certain  length,  and  least  intense  in  the  intermediate  state 
of  the  interfering  portions  ;  and  this  length  is  different  for 
light  of  different  colors." 

I  have  already  shown  in  detail  the  sufficiency  of  this  law 
for  explaining  all  the  phenomena  described  in  the  second 
and  third  books  of  Newton's  Optics,  as  well  as  some  others 
not  mentioned  by  Newton.  But  it  is  still  more  satisfactory 
to  observe  its  conformity  to  other  facts,  which  constitute  new 
and  distinct  classes  of  phenomena,  and  which  could  scarcely 
have  agreed  so  well  with  any  anterior  law,  if  that  law  had 
been  erroneous  or  imaginary  :  these  are  the  colors  of  fibres 
and  the  colors  of  mixed  plates. 

As  I  was  observing  the  appearance  of  the  fine  parallel  lines' 
of  light  which  are  seen  upon  the  margin  of  an  object  held  near 

*From  the  Philosophical  Transactions  for  1802,  p.  387. 


ERSITY 
MEMOIRS    OK    THE    WAVE-THEORY    OF    LIGHT 

the  eye,  so  as  to  intercept  the  greater  part  of  the  light  of  a 
distant  luminous  object,  and  which  are  produced  by  the  fringes 
caused  by  the  inflection  of  light  already  known,  I  observed 
that  they  were  sometimes  accompanied  by  colored  fringes, 
much  broader  and  more  distinct ;  and  I  soon  found  that 
these  broader  fringes  were  occasioned  by  the  accidental  inter- 
position of  a  hair.  .  In  order  to  make  them  more  distinct,  I 
employed  a  horse-hair,  but  they  were  then  no  longer  visible. 
\\rith  a  fibre  of  wool,  on  the  contrary,  they  became  very  large 
ard  conspicuous;  and,  with  a  single  silk -worm's  thread, 
their  magnitude  was  so  much  increased  that  two  or  three  of 
them  seemed  to  occupy  the  whole  field  of  view.  They  ap- 
peared to  extend  on  each  side  of  the  candle,  in  the  same  order 
as  the  colors  of  thin  plates  seen  by  transmitted  light.  It  oc- 
curred to  me  that  their  cause  must  be  sought  in  the  interfer- 
ence of  two  portions  of  light,  one  reflected  from  the  fibre,  the 
other  bending  round  its  opposite  side,  and  at  last  coinciding 
nearly  in  direction  with  the  former  portion  ;  that,  accordingly, 
as  both  portions  deviated  more  from  a  rectilinear  direction,  the 
difference  of  the  length  of  their  paths  would  become  gradual- 
ly greater  and  greater,  and  would  consequently  produce  the 
appearances  of  color  usual  in  such  cases ;  that  supposing 
them  to  be  inflected  at  right  angles,  the  difference  would 
amount  nearly  to  the  diameter  of  the  fibre,  and  that  this  dif- 
ference must  consequently  be  smaller  as  the  fibre  became 
smaller  ;  and,  the  number  of  fringes  in  a  right  angle  becoming 
smaller,  that  their  angular  distances  would  consequently  be- 
come greater,  and  the  whole  appearance  would  be  dilated.  It 
was  easy  to  calculate  that  for  the  light  least  inflected  the 
difference  of  the  paths  would  be  to  the  diameter  of  the  fibre 
very  nearly  as  the  deviation  of  the  ray  at  any  point  from  the 
rectilinear  direction  to  its  distance  from  the  fibre. 

I  therefore  made  a  rectangular  hole  in  a  card,  and  bent  its 
ends  so  as  to  support  a  hair  parallel  to  the  sides  of  the  hole  ; 
then,  upon  applying  the  eye  near  the  hole,  the  hair,  of  course, 
appeared  dilated  by  indistinct  vision  into  a  surface,  of  which 
the  breadth  was  determined  by  the  distance  of  the  hair  and 
the  magnitude  of  the  hole,  independently  of  the  temporary 
aperture  of  the  pupil.  When  the  hair  approached  so  near  to 
the  direction  of  the  margin  of  a  candle  that  the  inflected  light 
was  sufficiently  copious  to  produce  a  sensible  effect,  the  fringes 

63 


M  E  M  O I R  S    ON 

began  to  appear ;  and  it  was  easy  to  estimate  the  proportion 
of  their  breadth  to  the  apparent  breadth  of  the  hair  across  the 
image  of  which  they  extended.  I  found  that  six  of  the  bright- 
est red  fringes,  nearly  at  equal  distances,  occupied  the  whole 
of  that  image.  The  breadth  of  the  aperture  was  T{Hb->  and  its 
distance  from  the  hair  -f$  of  an  inch  ;  the  diameter  of  the  hair 
was  less  than  -g-J-g-  of  an  inch  ;  as  nearly  as  I  could  ascertain 
it  was  ¥^.  Hence,  we  have  y-j-J^  for  the  deviation  of  the 
first  red  fringe  at  the  distance  ^  ;  and  as  ^  :  T^-Q  :  :  -g-g-g-  • 
rro  O-OTT*  or  OTST  f°r  tne  difference  of  the  routes  of  the  rea 
light  where  it  was  most  intense.  The  measure  deduced  from 
Newton's  experiments  is  36|00.  I  thought  this  coincidence, 
with  only  an  error  of  one-ninth  of  so  minute  a  quantity,  suffi- 
ciently perfect  to  warrant  completely  the  explanation  of  the 
phenomenon,  and  even  to  render  a  repetition  of  the  experi- 
ment unnecessary  ;  for  there  are  several  circumstances  whicft 
make  it  difficult  to  calculate  much  more  precisely  what  ought 
to  be  the  result  of  the  measurement. 

When- a  number  of  fibres  of  the  same  kind — for  instance,  a 
uniform  lock  of  wool — are  held  near  to  the  eye,  we  see  an  ap- 
pearance of  halos  surrounding  a  distant  candle ;  but  their 
brilliancy,  and  even  their  existence,  depends  on  the  uniformity 
of  the  dimensions  of  the  fibres  ;  and  they  are  larger  as  the 
fibres  are  smaller.  It  is  obvious  that  they  are  the  immediate 
consequences  of  the  coincidence  of  a  number  of  fringes  of  the 
same  size,  which,  as  the  fibres  are  arranged  in  all  imaginable 
directions,  must  necessarily  surround  the  luminous  object  at 
equal  distances  on  all  sides,  and  constitute  circular  fringes. 

There  can  be  little  doubt  that  the  colored  atmospherical 
halos  are  of  the  same  kind  ;  their  appearance  must  depend  on 
the  existence  of  a  number  of  particles  of  water  of  equal  dimen- 
sions, and  in  a  proper  position  with  respect  to  the  luminary 
and  to  the  eye.  As  there  is  no  natural  limit  to  the  magnitude 
of  the  spherules  of  water,  we  may  expect  these  halos  to  vary 
without  limit  in  their  diameters,  and  accordingly  Mr.  Jordan 
has  observed  that  their  dimensions  are  exceedingly  various, 
and  has  remarked  that  they  frequently  change  during  the  time 
of  observation. 

I  first  noticed  the  colors  of  mixed  plates  in  looking  at  a 
candle  through  two  pieces  of  plate-glass  with  a  little  moisture 
between  them.  I  observed  an  appearance  of  fringes  resembling 

64: 


THE    WAVE-THEORY    OF    LIGHT 

the  common  colors  of  thin  plates  ;  and,  upon  looking  for  the 
fringes  by  reflection,  I  found  that  these  new  fringes  were 
always  in  the  same  direction  as  the  other  fringes,  but  many 
times  larger.  By  examining  the  glasses  with  a  magnifier,  I 
perceived  that  wherever  these  fringes  were  visible  the  moist- 
ure was  intermixed  with  portions  of  air,  producing  an  appear- 
ance similar  to  dew.  I  then  supposed  that  the  origin  of  the 
colors  was  the  same  as  that  of  the  colors  of  halos  ;  but,  on 
a  more  minute  examination,  I  found  that  the  magnitude  of  the 
portions  of  air  and  water  was  by  no  means  uniform,  and  that 
the  explanation  was,  therefore,  inadmissible.  It  was,  however, 
easy  to  find  two  portions  of  light  sufficient  for  the  production 
of  these  fringes  ;  for  the  light  transmitted  through  the  water, 
moving  in  it  with  a  velocity  different  from  that  of  the  light 
passing  through  the  interstices  filled  only  with  air,  the  two 
portions  would  interfere  with  each  other  and  produce  effects 
of  color  according  to  the  general  law.  The  ratio  of  the 
velocities  in  water  and  in  air  is  that  of  3  to  4 ;  the  fringes 
ought,  therefore,  to  appear  where  the  thickness  is  six  times  as 
great  as  that  which  corresponds  to  the  same  color  in  the  com- 
mon case  of  thin  plates ;  and,  upon  making  the  experiment 
with  a  plane  glass  and  a  lens  slightly  convex,  I  found  the  sixtli 
dark  circle  actually  of  the  same  diameter  as  the  first  in  the 
new  fringes.  The  colors  are  also  very  easily  produced  when 
butter  or  tallow  is  substituted  for  water;  and  the  rings  then 
become  smaller,  on  account  of  the  greater  refractive  density 
of  the  oils ;  but  when  water  is  added,  so  as  to  fill  up  the  in- 
terstices of  the  oil,  the  rings  are  very  much  enlarged ;  for  here 
the  difference  only  of  the  velocities  in  water  and  in  oil  is  to 
be  considered,  and  this  is  much  smaller  than  the  difference 
between  air  and  water.  All  these  circumstances  are  sufficient 
to  satisfy  us  with  respect  to  the  truth  of  the  explanation  ;  and 
it  is  still  more  confirmed  by  the  effect  of  inclining  the  plates 
to  the  direction  of  the  light;  for  then,  instead  of  dilating,  like 
the  colors  of  thin  plates,  these  rings  contract:  and  this  is  the 
obvious  consequence  of  an  increase  of  the  length  of  the  paths 
of  light,  which  now  traverse  both  mediums  obliquely ;  and  the 
effect  is  everywhere  the  same  as  that  of  a  thicker  plate. 

It  must,  however,  be  observed  that  the  colors  are  not  pro- 
duced  in  the   whole   light  that   is    transmitted   through   the 
mediums  :  a  small  portion  only  of  each  pencil,  passing  through 
E  65 


MEMOIRS    ON 

the  water  contiguous  to  the  edges  of  the  particle,  is  sufficient- 
ly coincident  with  the  light  transmitted  by  the  neighboring 
portions  of  air  to  produce  the  necessary  interference  ;  and  it 
is  easy  to  show  that,  on  account  of  the  natural  concavity  of  the 
surface  of  each  portion  of  the  fluid  adhering  to  the  two  pieces 
of  glass,  a  considerable  portion  of  the  light  which  is  beginning 
to  pass  through  the  water  will  be  dissipated  laterally  by  re- 
flection at  its  entrance,  and  that  much  of  the  light  passing 
through  the  air  will  be  scattered  by  refraction  at  the  second 
surface.  For  these  reasons  the  fringes  are  seen  when  the 
plates  are  not  directly  interposed  between  the  eye  and  the 
luminous  object;  and  on  account  of  the  absence  of  foreign 
light,  even  more  distinctly  than  when  they  are  in  the  same 
right  line  with  that  object.  And  if  we  remove  the  plates  to  a 
considerable  distance  out  of  this  line,  the  rings  are  still  visible 
and  become  larger  than  before  ;  for  here  the  actual  route  of 
the  light  passing  through  the  air  is  longer  than  that  of  the 
light  passing  more  obliquely  through  the  water,  and  the  differ- 
ence in  the  times  of  passage  is  lessened.  It  is?<  however,  im- 
possible to  be  quite  confident  with  respect  to  the  causes  of 
these  minute  variations,  without  some  means  of  ascertaining 
accurately  the  forms  of  the  dissipating  surfaces. 

In  applying  the  general  law  of  interference  to  these  colors, 
as  well  as  to  those  of  thin  plates  already  known,  I  must  con- 
fess that  it  is  impossible  to  avoid  another  supposition,  which  is 
a  part  of  the  undulatory  theory — that  is,  that  the  velocity  of 
light  is  the  greater  the  rarer  the  medium  ;  and  that  there  is 
also  a  condition  annexed  to  the  explanation  of  the  colors  of 
thin  plates  which  involves  another  part  of  the  same  theory— 
that  is,  that  where  one  of  the  portions  of  light  has  been  re- 
flected at  the  surface  of  a  rarer  medium,  it  must  be  supposed 
to  be  retarded  one-half  of  the  appropriate  interval — for  in- 
stance, in  the  central  black  spot  of  a  soap-bubble,  where  the 
actual  lengths  of  the  paths  very  nearly  coincide,  but  the  effect 
is  the  same  as  if  one  of  the  portions  had  been  so  retarded  as  to 
destroy  the  other.  From  considering  the  nature  of  this  cir- 
cumstance,- I  ventured  to  predict  that  if  the  two  reflections 
were  of  the  same  kind,  made  at  the  surfaces  of  a  thin  plate  of 
a  density  intermediate  between  the  densities  of  the  mediums 
containing  it,  the  effect  would  be  reversed,  and  the  central 
spot,  instead  of  black,  would  become  white ;  and  I  have  now 

66 


THE    WAVE-THEORY    OF    LIGHT 

the  pleasure  of  stating  that  I  have  fully  verified  this  predic- 
tion by  interposing  a  drop  of  oil  of  sassafras  between  a  prism 
of  fiint-glass  and  a  lens  of  crown-glass ;  the  central  spot  seen 
by  reflected  light  was  white  and  surrounded  by  a  dark  ring. 
It  was,  however,  necessary  to  use  some  force  in  order  to  pro- 
duce a  contact  sufficiently  intimate  ;  and  the  white  spot  dif- 
fered, even  at  last,  in  the  same  degree  from  perfect  whiteness 
as  the  black  spot  usually  does  from  perfect  blackness. 

[Three  pages  of  speculation    concerning  dispersion   are  here 
omitted.] 

67 


EXPEEIMENTS  AND  CALCULATIONS  REL- 
ATIVE TO  PHYSICAL  OPTICS* 

A  BAKEKIAN   LECTUEE 

Read  Noveniber  24,  1803 


I.    EXPERIMENTAL    DEMONSTRATION    OF   THE  GENERAL  LAW  OF 
THE    INTERFERENCE    OF    LIGHT. 

IN  making  some  experiments  on  the  fringes  of  colors  ac- 
companying shadows,  I  have  found  so  simple  and  so  demon- 
strative a  proof  of  the  general  law  of  the  interference  of  two 
portions  of  light,  which  I  have  already  endeavored  to  estab- 
lish, that  I  think  it  right  to  lay  before  the  Royal  Society  a 
short  statement  of  the  facts  which  appear  to  me  so  decisive. 
The  proposition  on  which  I  mean  to  insist  at  present  is  simply 
this — that  fringes  of  colors  are  produced  by  the  interference 
of  two  portions  of  light ;  and  I  think  it  will  not  be  defied  by 
the  most  prejudiced  that  the  assertion  is  proved  by  the  ex- 
periments I  am  about  to  relate,  which  may  be  repeated  with 
great  ease  whenever  the  sun  shines,  and  without  any  other  ap- 
paratus than  is  at  hand  to  every  one. 

Experiment  1.  I  made  a  small  hole  in  a  window-shutter,  and 
covered  it  with  a  piece  of  thick  paper,  which  I  perforated  with 
a  fine  needle.  For  greater  convenience  of  observation  I  placed 
a  small  looking-glass  without  the  window-shutter,  in  such  a 
position  as  to  reflect  the  sun's  light  in  a  direction  nearly  hor- 
izontal upon  the  opposite  wall,  and  to  cause  the  cone  of  di- 
verging light  to  pass  over  a  table  on  which  were  several  little 
screens  of  card-paper.  I  brought  into  the  sunbeam  a  slip  of 

*From  the  Philosophical  Transactions  for  1804. 


MEMOIRS    ON    THE    WAVE-THEORY    OF    LIGHT 

card  about  one-thirtieth  of  an  inch  in  breadth,  and  observed 
its  shadow,  either  on  the  wall  or  on  other  cards  held  at  differ- 
ent distances.  Besides  the  fringes  of  color  on  each  side  of 
the  shadow,  the  shadow  itself  was  divided  by  similar  parallel 
fringes  of  smaller  dimensions,  differing  in  number  according 
to  the  distance  at  which  the  shadow  was  observed,  but  leaving 
the  middle  of  the-'  shadow  always  white.  Now  these  fringes 
were  the  joint  effects  of  the  portions  of  light  passing  on  each 
side  of  the  slip  of  card,  and  inflected,  or  rather  diffracted,  into 
the  shadow  ;  for  a  little  screen  being  placed  a  few  inches 
from  the  card  so  as  to  receive  either  edge  of  the  shadow  on 
its  margin,  all  the  fringes  which  had  before  been  observed  in 
the  shadow  on  the  wall  immediately  disappeared,  although  the 
light  inflected  on  the  other  side  was  allowed  to  retain  its  course, 
and  although  this  light  must  have  undergone  any  modification 
that  the  proximity  of  the  other  edge  of  the  slip  of  card  might 
have  been  capable  of  occasioning.  When  the  interposed  screen 
was  more  remote  from  the  narrow  card,  it  was  necessary  to 
plunge  it  more  deeply  into  the  shadow,  in  order  to  extinguish 
the  parallel  lines  ;  for  here  the  light  diffracted  from  the  edge 
of  the  object  had  entered  farther  into  the  shadow  in  its  way 
towards  the  fringes.  Nor  was  it  for  want  of  a  sufficient  in- 
tensity of  light  that  one  of  the  two  portions  was  incapable  of 
producing  the  fringes  alone ;  for  when  they  were  both  unin- 
terrupted, the  lines  appeared,  even  if  the  intensity  was  reduced 
to  one- tenth  or  one-twentieth. 

Experiment  2.  The  crested  fringes  described  by  the  ingenious 
and  accurate  Grimaldi  afford  an  elegant  variation  of  the  pre- 
ceding experiment  and  an  interesting  example  of  a  calcula- 
tion grounded  on  it.  When  a  shadow  is  formed  by  an  object 
which  has  a  rectangular  termination  besides  the  usual  external 
fringes  there  are  two  or  three  alternations  of  colors,  beginning 
from  the  line  which  bisects  the  angle,  disposed  on  each  side  of 
it  in  curves,  which  are  convex  towards  the  bisecting  line,  and 
which  converge  in  some  degree  towards  it  as  they  become 
more  remote  from  the  angular  point.  These  fringes  are  also 
the  joint  effect  of  the  light  which  is  inflected  directly  towards 
the  shadow  from  each  of  the  two  outlines  of  the  object  ;  for  if 
a  screen  be  placed  within  a  few  inches  of  the  object,  so  as  to 
receive  only  one  of  the  edges  of  the  shadow,  the  whole  of  the 
fringes  disappear  ;  if,  on  the  contrary,  the  rectangular  point 


MEM  OIKS    ON 

of  the  screen  be  opposed  to  the  point  of  the  shadow  so  as 
barely  to  receive  the  angle  of  the  shadow  on  its  extremity,  the 
fringes  will  remain  undisturbed. 

II.    COMPARISON     OF    MEASURES    DEDUCED    FROM    VARIOUS    EX- 
PERIMENTS. 

If  we  now  proceed  to  examine  the  dimensions  of  the  fringes 
under  different  circumstances,  we  may  calculate  the  differences 
of  the  lengths  of  the  paths  described  by  the  portions  of  light 
which  have  thus  been  proved  to  be  concerned  in  producing 
those  fringes  ;  and  we  shall  find  that  where  the  lengths  are 
equal  the  light  always  remains  white ;  but  that  where  either 
the  brightest  light  or  the  light  of  any  given  color  disappears 
and  reappears  a  first,  a  second,  or  a  third  time,  the  differences 
of  the  lengths  of  the  paths  of  the  two  portions  are  in  arithmet- 
ical progression,  as  nearly  as  we, can  expect  experiments  of  this 
kind  to  agree  with  each  other.  I  shall  compare,  in  this  point 
of  view,  the  measures  deduced  from  several  experiments  of 
Newton  and  from  some  of  my  own. 

In  the  eighth  and  ninth  observations  of  the  third  book  of 
Newton's  Optics  some  experiments  are  related  which,  together 
with  the  third  observation,  will  furnish  us  with  the  data  neces- 
sary for  the  calculation.  Two  knives  were  placed,  with  their 
edges  meeting  at  a  very  acute  angle,  in  a  beam  of  the  sun's 
light,  admitted  through  a  small  aperture,  and  the  point  of  con- 
course of  the  two  first  dark  lines  bordering  the  shadows  of  the  re- 
spective knives  was  observed  at  various  distances.  The  results 
of  six  observations  are  expressed  in  the  first  three  lines  of  the 
first  table.  On  the  supposition  that  the  dark  line  is  produced 
by  the  first  interference  of  the  light  reflected  from  the  edges  of 
the  knives,  with  the  light  passing  in  a  straight  line  between 
them,  we  may  assign,  by  calculating  the  difference  of  the  two 
paths,  the  interval  for  the  first  disappearance  of  the  brightest 
light,  as  it  is  expressed  in  the  fourth  line.  The  second  table 
contains  the  results  of  a  similar  calculation  from  Newton's  ob- 
servations on  the  shadow  of  a  hair  ;  and  the  third,  from  some 
experiments  of  my  own  of  the  same  nature  ;  the  second  bright 
line  being  supposed  to  correspond  to  a  double  interval,  the  sec- 
ond dark  line  to  a  triple  interval,  and  the  succeeding  lines  to 
depend  on  a  continuation  of  the  progression.  The  unit  of  all 
,the  tables  is  an  inch. 

70 


THE    WAVE-THEORY    OF    LIGHT 

TABLE  I. — Observation  9.     N. 

Distance  of  the  knives  from  the  aperture 101 

Distance  of  the 

paper  from 

the  knives 1£  ^  &t  32  96  131 

Distance  b  e  - 

tween    the 

edges  of  the 

knives    o  p- 

posite  to  the 

point    of 

concourse 012  .020  .034  .057  .081  .087 

Interval  of  dis- 
appearance  0000122  .0000155  .0000182  .0000167  .0000166  .0000166 

TABLE  II.— Observation  3.     N. 

Breadth  of  the  hair ^ 

Distance  of  the  hair  from  the  aperture 144 

Distances  of  the  scale  from  the  aperture 150  252 

(Breadths of  the  shadow ^  £) 

Breadth  between  ihe  second  pair  of  bright  lines -g?  T4T 

Interval  of  disappearance,  or  half  the  difference  of  the 

paths 0000151  .0000173 

Breadth  between  the  third  pair  of  bright  lines . ..  ^  T3¥ 

Interval  of  disappearance,  one-fourth  of  the  difference..   .0000130  .0000143 

TABLE  III. — Experiment  3. 

Breadth  of  the  object 434 

Distance  of  the  object  from  the  aperture 125 

Distance  of  the  wall  from  the  aperture 250 

Distance  of  the  second  pair  of  dark  lines  from  each  other 1.167 

Interval  of  disappearance,  one-third  of  the  difference 0000149 

Experiment  4. 

Breadth  of  the  wire 083 

Distance  of  the  wire  from  the  aperture 32 

Distance  of  the  wall  from  the  aperture 250 

(Breadth  of  the  shadow,  by  three 

measurements 815.  .826,  or  .827  ;  mean,  .823) 

Distance  of  the  first  pair  of  dark  lines  1.165,  1.170,  or  1.160;  mean,  1.165 

Interval  of  disappearance  .......... 0000194 

Distance  of  the  second  pair  of  dark 

lines 1.402.  1.395,  or  1.400;  mean.  1.399 

Interval  of  disappearance 0000137 

Distance  of  the  third  pair  of  dark 

lines. , 1.594,  1.580,  or  1.585 ;  mean,  1.586 

Interval  of  disappearance 0000128 

71 


MEMOIRS    ON 

It  appears,  from  five  of  the  six  observations  of  the  first 
table,  in  which  the  distance  of  the  shadow  was  varied  from 
about  3  inches  to  11  feet,  and  the  breadth  of  the  fringes  was 
increased  in  the  ratio  of  7  to  1,  that  the  difference  of  the 
routes  constituting  the  interval  of  disappearance  varied  but 
one-eleven tli  at  most ;  and  that  in  three  out  of  the  five  it 
agreed  with  the  mean,  either  exactly  or  within  y^-  part. 
Hence  we  are  warranted  in  inferring  that  the  interval  appro- 
priate to  the  extinction  of  the  brightest  light  is  either  accu- 
rately or  very  nearly  constant. 

But  it  may  be  inferred  from  a  comparison  of  all  the  other 
observations  that  when  the  obliquity  of  the  reflection  is  very 
great  some  circumstance  takes  place  which  causes  the  inter- 
val thus  calculated  to  be  somewhat  greater ;  thus,  in  the  elev- 
enth line  of  the  third  table  it  comes  out  one-sixth  greater  than 
the  mean  of  the  five  already  mentioned.  On  the  other  hand, 
the  mean  of  two  of  Newton's  experiments  and  one  of  mine  is 
a  result  about  one-fourth  less  than  the  former.  With  respect 
to  the  nature  of  this  circumstance  I  cannot  at  present  form  a 
decided  opinion ;  but  I  conjecture  that  it  is  a  deviation  of 
some  of  the  light  concerned,  from  the  rectilinear  direction  as- 
signed to  it,  arising  either  from  its  natural  diffraction,  by  which 
the  magnitude  of  the  shadow  is  also  enlarged,  or  from  some 
other  unknown  cause.  If  we  imagined  the  shadow  of  the 
wire  and  the  fringes  nearest  it  to  be  so  contracted  that  the 
motion  of  the  light  bounding  the  shadow  might  be  rectilinear, 
we  should  thus  make  a  sufficient  compensation  for  this  devia- 
tion ;  but  it  is  difficult  to  point  out  what  precise  track  of  the 
light  would  cause  it  to  require  this  correction. 

The  mean  of  the  three  experiments  which  appear  to  have  been 
least  affected  by  this  unknown  deviation  gives  .0000127  for  the 
interval  appropriate  to  the  disappearance  of  the  brightest 
light ;  and.  it  may  be  inferred  that  if  they  had  been  wholly  ex- 
empted from  its  effects  the  measure  would  have  been  some- 
what smaller.  Now  the  analogous  interval,  deduced  from  the 
experiments  of  Newton  on  this  plate,  is  .0000112,  which  is 
about  one-eighth  less  than  the  former  result ;  and  this  appears 
to  be  a  coincidence  fully  sufficient  to  authorize  us  to  attribute 
these  two  classes  of  phenomena  to  the  same  cause.  It  is  very 
easily  shown,  with  respect  to  the  colors  of  thin  plates,  that 
each  kind  of  light  disappears  and  reappears  where  the  differ- 

72 


THE    WAVE-THEORY    OF    LIGHT 

ences  of  the  routes  of  two  of  its  portions  are  in  arithmetical 
progression*}  and  we  have  seen  that  the  same  law  may  be  in 
general  inferred  from  the  phenomena  of  diffracted  light,  even 
independently  of  the  analogy. 

The  distribution  of  the  colors  is  also  so  similar  in  both  cases 
as  to  point  immediately  to  a  similarity  in  the  causes.  In  the 
thirteenth  observation  of  the  second  part  of  the  first  book 
Newton  relates  that  the  interval  of  the  glasses  where  the  rings 
appeared  in  red  light  was  to  the  interval  where  they  appeared  in 
violet  light  as  14  to  9  ;  and,  in  the  eleventh  observation  of  the 
third  book,  that  the  distances  between  the  fringes,  under  the 
same  circumstances,  were  the  twenty-second  and  the  twenty-sev- 
enth of  an  inch.  Hence,  deducting  the  breadth  of  the  hair  and 
taking  the  squares,  in  order  to  find  the  relation  of  the  difference 
of  the  routes,  we  have  the  proportion  of  14  to  9£,  which  scarcely 
differs  from  the  proportion  observed  in  the  colors  of  the  thin 
plate. 

We  may  readily  determine  from  this  general  principle  the 
form  of  the  crested  fringes  of  Grimaldi,  already  described }  for 
it  will  appear  that,  under  the  circumstances  of  the  experiment 
related,  the  points  in  which  the  differences  of  the  lengths  of 
the  paths  described  by  the  two  portions  of  light  are  equal  to 
a  constant  quantity,  and  in  which,  therefore,  the  same  kinds 
of  light  ought  to  appear  or  disappear,  are  always  found  in 
equilateral  hyperbolas,  of  which  the  axes  coincide  with  the 
outlines  of  the  shadow,  and  the  asymptotes  nearly  with  the 
diagonal  line.  Such,  therefore,  must  be  the  direction  of  the 
fringes  ;  and  this  conclusion  agrees  perfectly  with  the  observa- 
tion. But  it  must  be  remarked  that  the  parts  near  the  out- 
lines of  the  shadow  are  so  much  shaded  off  as  to  render  the 
character  of  the  curve  somewhat  less  decidedly  marked  where 
it  approaches  to  its  axis.  These  fringes  have  a  slight  resem- 
blance to  the  hyperbolic  fringes  observed  by  Newton ;  but  the 
analogy  is  only  distant. 

[///.  Application  to  the  Supernumerary  Rainbows,  omitted.] 

IV.     ARGUMENTATIVE     INFERENCE     RESPECTING     THE     NATURE 

OF    LIGHT. 

The  experiment  of  Grimaldi  on  the  crested  fringes  within 
the  shadow,  together  with  several  others  of  his  observations 

73 


MEMOIRS    ON 

equally  important,  has  been  left  unnoticed  by  Newton.  Those 
who  are  attached  to  the  Newtonian  theory  of  light,  or  to  the 
hypothesis  of  modern  opticians  founded  on  views  still  less 
enlarged,  would  do  well  to  endeavor  to  imagine  anything  like 
an  explanation  of  these  experiments  derived  from  their  own 
doctrines  ;  and  if  they  fail  in  the  attempt,  to  refrain  at  least 
from  idle  declamation  against  a  system  which  is  founded  on 
the  accuracy  of  its  application  to  all  these  facts,  and  to  a  thou- 
sand others  of  a  similar  nature. 

From  the  experiments  and  calculation  which  have  been  pre- 
mised, we  may  be  allowed  to  infer  that  homogeneous  light 
at  certain  equal  distances  in  the  direction  of  its  motion  is  pos- 
sessed of  opposite  qualities  capable  of  neutralizing  or  destroy- 
ing each  other,  and  of  extinguishing  the  light  where  they 
happen  to  be  united;  that  these  qualities  succeed  each  other 
alternately  in  successive  concentric  superficies,  at  distances 
which  are  constant  for  the  same  light  passing  through  the 
same  medium.  From  the  agreement  of  the  measures,  and  from 
the  similarity  of  the  phenomena,  we  may  conclude  that  these 
intervals  are  the  same  as  are  concerned  in  the  production  of  the 
colors  of  thin  plates  ;  but  these  are  shown,  by  the  experi- 
ments of  Newton,  to  be  the  smaller  the  denser  the  medium; 
and  since  it  may  be  presumed  that  their  number  must  neces- 
sarily remain  unaltered  in  a  given  quantity  of  light,  it  follows, 
of  course,  that  light  moves  more  slowly  in  a  denser  than  in  a 
rarer  medium';  and  this  being  granted,  it  must  be  allowed  that 
refraction  is  not  the  effect  of  an  attractive  force  directed  to  a 
denser  medium.  The  advocates  for  the  projectile  hypothesis 
of  light  must  consider  which  link  in  this  chain  of  reasoning 
they  may  judge  to  be  the  most  feeble,  for  hitherto  I  have 
advanced  in  this  paper  no  general  hypothesis  whatever.  But 
since  we  know  that  sound  diverges  in  concentric  superficies, 
and  that  musical  sounds  consist  of  opposite  qualities,  capable 
of  neutralizing  each  other,  and  succeeding  at  certain  equal 
intervals,  which  are  different  according  to  the  difference  of 
the  note,  we  are  fully  authorized  to  conclude  that  there  must 
be  some  strong  resemblance  between  the  nature  of  sound  and 
that  of  light. 

I  have  not,  in  the  course  of  these  investigations,  found  any 
reason  to  suppose  the  presence  of  such  an  inflecting  medium 
in  the  neighborhood  of  dense  substances  as  I  was  formerly 

74 


THE    WAVE-T11EOKY    OF    LIGHT 

inclined  to  attribute  to  them  ;  and,  upon  considering  the  phe- 
nomena of  the  aberration  of  the  stars,  I  am  disposed  to  believe 
that  the  luminiferous  ether  pervades  the  substance  of  all  ma- 
terial bodies,  with  little  or  no  resistance,  as  freely,  perhaps,  as 
the  wind  passes  through  a  grove  of  trees. 

The  observations  on  the  effects  of  diffraction  and  inter- 
ference may,  perhaps,  sometimes  be  applied  to  a  practical  pur- 
pose in  making  us  cautious  in  our  conclusions  respecting  the 
appearances  of  minute  bodies  viewed  in  a  microscope.  The 
shadow  of  a  fibre,  however  opaque,  placed  in  a  pencil  of  light 
admitted  through  a  small  aperture,  is  always  somewhat  less  dark 
in  the  middle  of  its  breadth  than  in  the  parts  on  each  side.  A 
similar  effect  may  also  take  place,  in  some  degree,  with  respect 
to  the  image  on  the  retina,  and  impress  the  sense  with  an  idea 
of  a  transparency  which  has  no  real  existence  ;  and  if  a  small 
portion  of  light  be  really  transmitted  through  the  substance, 
this  may  again  be  destroyed  by  its  interference  with  the  dif- 
fracted light,  and  produce  an  appearance  of  partial  opacity, 
instead  of  uniform  semi-transparency.  Thus  a  central  dark  spot 
and  a  light  spot,  surrounded  by  a  darker  circle,  may  respec- 
tively be  produced  in  the  images  of  a  semi-transparent  and 
an  opaque  corpuscle,  and  impress  us  with  an  idea  of  a  com- 
plication of  structure  which  does  not  exist.  In  order  to  detect 
the  fallacy,  we  make  two  or  three  fibres  cross  each  other,  and 
view  a  number  of  globules  contiguous  to  each  other;  or  we 
may  obtain  a  still  more  effectual  remedy  by  changing  the  mag- 
nifying power;  and  then,  if  the  appearance  remain  constant  in 
kind  and  in  degree,  we  may  be  assured  that  it  truly  represents 
the  nature  of  the  substance  to  be  examined.  It  is  natural  to 
inquire  whether  or  not  the  figures  of  the  globules  of  blood 
delineated  by  Mr.  Hewson  in  the  Phil.  Trans.,  vol.  Ixiii.,  for 
1773,  might  not  in  some  measure  have  been  influenced  by  a 
deception  of  this  kind  ;  but,  as  far  as  I  have  hitherto  been  able 
to  examine  the  globules  with  a  lens  of  one-fiftieth  of  an  inch 
focus,  I  have  found  them  nearly  such  as  Mr.  Hewson  has  de- 
scribed them. 

[  V.  Remarks  on  the  Colors  of  Natural  Bodies,  omitted.] 

VI.    EXPERIMENT  ON  THE  DARK  RAYS  OF  RITTER 

Experiment  6.  The  existence  of  solar  rays  accompanying 
light,  more  refrangible  than  the  violet  rays  and  cognizable  by 

75 


MEMOIRS    ON 

their  chemical  effects,  was  first  ascertained  by  Mr.  Bitter  ;  but 
Dr.  Wollaston  made  the  same  experiments  a  very  short  time 
afterwards  without  having  been  informed  of  what  had  been 
done  on  the  Continent.  These  rays  appear  to  extend  beyond 
the  violet  rays  of  the  prismatic  spectrum,  through  a  space 
nearly  equal  to  that  which  is  occupied  by  the  violet.  In  order 
to  complete  the  comparison  of  their  properties  with  those  of 
visible  light,  I  was  desirous  of  examining  the  effect  of  their  re- 
flection from  a  thin  plate  of  air,  capable  of  producing  the  well- 
known  rings  of  colors.  For  this  purpose  I  formed  an  image 
of  the  rings,  by  means  of  the  solar  microscope,  with  the  appa- 
ratus which  I  have  described  in  the  Journals  of  the  Royal 
Institution,  and  I  threw  this  image  on  paper  dipped  in  a  solu- 
tion of  nitrate  of  silver,  placed  at  the  distance  of  about  nine 
inches  from  the  microscope.  In  the  course  of  an  hour  portions 
of  three  dark  rings  were  very  distinctly  visible,  much  smaller 
than  the  brightest  rings  of  the  colored  image,  and  coinciding 
very  nearly  in  their  dimensions  with  the  rings  of  violet  light 
that  appeared  upon  the  interposition  of  violet  glass.  I  thought 
the  dark  rings  were  a  little  smaller  than  the  violet  rings,  but 
the  difference  was  not  sufficiently  great  to  be  accurately  ascer- 
tained ;  it  might  be  as  much  as  fa  or  fa  of  the  diameters,  but 
not  greater.  It  is  the  less  surprising  that  the  difference  should 
be  so  small,  as  the  dimensions  of  the  colored  rings  do  not  by 
any  means  vary  at  the  violet  end  of  the  spectrum  so  rapidly  as 
at  the  red  end.  For  performing  this  experiment  with  very 
great  accuracy  a  heliostat  would  be  necessary,  since  the  motion 
of  the  sun  causes  a  slight  change  in  the  place  of  the  image  ; 
and  leather  impregnated  with  the  muriate  of  silver  would 
indicate  the  effect  with  greater  delicacy.  The  experiment, 
however,  in  its  present  state,  is  sufficient  to  complete  the  anal- 
ogy of  the  invisible  with  the  visible  rays,  and  to  show  that  they 
are  equally  liable  to  the  general  law  which  is  the  principal  sub- 
ject of  this  paper.  If  we  had  thermometers  sufficiently  delicate, 
it  is  probable  that  we  might  acquire,  by  similar  means,  infor- 
mation still  more  interesting  with  respect  to  the  rays  of  invis- 
ible heat  discovered  by  Dr.  Herschel ;  but  at  present  there  is 
great  reason  to  doubt  of  the  practicability  of  such  an  experi- 
ment. 

76 


THE    WAVE-THEORY    OF    LIGHT 


BIOGRAPHICAL  SKETCH 

THOMAS  YOUNG  was  born  at  Milverton.  England,  in  1773, 
and  died  at  London  in  1829.  His  education,  in  respect  to  the 
amount  of  ground  it  co.vered,  is  quite  as  remarkable  as  his  later 
scientific  work.  As  a  lad  he  showed  marked  proficiency  in 
linguistic  studies,  acquired  great  mechanical  skill,  distin- 
guished himself  in  drawing,  music,  and  athletics.  As  a  young 
man  he  pursued  his  university  studies  at  London,  Edinburgh, 
Gb'ttingeu,  and  Cambridge. 

The  following  programme  of  his  daily  work  at  Gottingen  in 
the  autumn  of  1795  characterizes  at  once  the  lad,  the  youth, 
and  the  mature  man: 

"  At  8,  I  attend  Spittler's  course  on  the  History  of  the  Principal  States 
of  Europe,  exclusive  of  Germany. 

"  At  9,  Arnemann  on  Materia  Medica. 

"At  10,  Richter  on  Acute  Diseases. 

"  At  11,  twice  a  week,  private  lessons  from  Blessman,  the  academical 
dancing-master. 

"At  12,  I  dine  at  Ruhlander's  table  d'hote. 

"At  1,  twice  a  week,  lessons  on  the  clavichord  from  Forkel;  and  twice 
a  week  at  home,  from  Fiorillo  on  Drawing. 

"  At  2,  Lichtenberg  on  Physics. 

"At  3,  I  ride  in  the  academical  manege,  under  the  instruction  of  Ayrer, 
four  times  a  week. 

"  At  4,  Stromeyer  on  Diseases. 

*'  At  5,  Blumenbach  on  Natural  History. 

"At  6,  twice  Blessman  with  other  pupils,  and  twice  Forkel." 

He  was  born  of  a  well-to-do  Quaker  family;  he  inherited 
ample  money ;  he  had  all  that  travel,  leisure,  and  good  society 
could  do  for  a  man.  Only  in  one  particular  does  his  education 
appear  to  have  been  defective — viz.,  in  the  absence  of  any 
training  in  advanced  dynamics  or  in  higher  mathematical 
analysis. 

In  1800  he  completed  his  medical  studies  at  Cambridge,  and 
settled  as  a  practising  physician  in  London.  In  the  year  fol- 
lowing he  was  appointed  to  the  professorship  of  natural  philos- 
ophy in  the  then  newly  founded  Royal  Institution,  a  position 
from  which  he  resigned  at  the  end  of  two  years  in  order  to 
devote  himself  more  completely  to  the  practice  of  medicine. 
It  was  during  his  occupancy  of  this  chair  that  he  published  the 

77 


MEMOIRS    ON    THE    WAVE-THEORY    OF    LIGHT 

three  papers  reprinted  in  this  volume,  the  first  of  which  is  pos- 
sibly the  most  important  of  his  contributions  to  physics.  It 
was  during  this  period  also  that  he  wrote  his  Lectures  on  Natural 
Philosophy,  which  must  always  be  reckoned  as  a  potent  factor 
in  the  spread  of  sound  physical  science  in  the  nineteenth  cen- 
tury, while  its  bibliography  of  more  than  four  hundred  quarto 
pages  is  to-day  valuable  as  well  as  classic. 

But  nothing  short  of  a  catalogue  of  his  papers  can  give  one 
an  adequate  idea  of  the  varied  activity  of  this  man  during  the 
remaining  quarter-century  of  his  life.  His  contributions  cover 
fields  as  diverse  as  the  physiology  of  the  human  eye,  hydro- 
dynamics, music,  paleography,  atmospheric  refraction,  theory 
of  tides,  tables  of  mortality,  theory  of  structures.  His  expla- 
nation of  color-vision  as  due  to  the  presence  of  three  sets  of 
nerve  fibres  in  the  retina,  which,  when  excited,  give  respectively 
sensations  of  red,  green,  and  violet,  has  been  adopted  and  modi- 
fied by  Helmholtz,  and  is  to-day  perhaps  the  most  widely 
accepted  of  the  various  theories  on  this  subject. 

After  all,  it  rnnst^  be  confessed,  even  by  his  most  ardent 
admirers,  that  Young's  style  is,  in  general,  far  from  clear. 
Whether  this  is  in  any  way  connected  with  his  lack  of  mathe- 
matical training,  or  whether  it  is  due  to  the  fact  that  his  own 
clear  intuitions  bridged  most  of  the  gaps  in  his  written  work, 
it  is  difficult  to  say ;  but  in  any  event  many  of  his  papers  are 
obscure,  and  few  of  them  are  read.  The  reader  who  desires  a 
full  biography  will  find  it  in  Dr.  Peacock's  Life  of  Young 
(London,  1855).  This  biographer  also  edited  his  Miscellaneous 
Works,  3  vols.  (London,  1855).  All  his  papers,  however,  which 
are  of  especial  interest  to  the  student  of  physics  are  contained 
in  the  lectures  on  Natural  Philosophy  (London,  1807). 

78 


MEMOIR  ON  THE  DIFFRACTION  OF  LIGHT 

o 
"  CROWNED  "    BY    THE   FRENCH    ACADEMY    OF   SCIENCES    IN    1819 

BY 

A.  FRE8NEL 

Natura  simplex  et  fecunda. 

-* 

MEMOIR   ON    THE   ACTION  OF   RAYS   OF 
POLARIZED  LIGHT  UPON  EACH  OTHER 

4. 
V 

BY 

MESSRS.  ARAGO  AND  FRESNEL 

(Annales  de  Ghimie  et  cto  ffysique.  t.  x.,  p.  288,  1819.) 
79 


CONTENTS 

PAGE 

On  tlie  Insufficiency  of  the  Corpuscular  Theory  and  of  Young's  Views 
concerning  Interference 81 

PJienomena  of  Diffraction  explained  by  the  Combination  of  Huygentfs 

Principle  with  that  of  Interference 108 

On  the  Interference  of  Polarized  Light ; .  145 


80 


FRESNEL'S  PRIZE  MEMOIR  ON  THE  DIP- 
FRACTION  OF  LIGHT 


[The  Introduction,  covering  fourteen  pages  and  describing  in 
the  most  general,  way  the  defects  of  the  emission-theory  and  some 
of  the  merits  of  the  wave-theory,  is  omitted.] 

SECTION   I 

11.  It  might  appear  that  on  the  emission -theory  nothing 
would  be  simpler  than  the  phenomena  of  shadows,  especially 
when  the  source  of  light  is  merely  a  point ;  but,  on  the  con- 
trary, nothing  is  more  complicated.  If  we  suppose  the  surface 
of  the  body  producing  the  shadow  to  be  endowed  with  a  re- 
pulsive property  capable  \>f  changing  the  direction  of  rays  of 
light  passing  very  near  it,  we  should  then  expect  only  to  see 
the  shadows  increase  in  size  and,  towards  their  edges,  to  blend 
a  little  with  the  illuminated  area;  while,  as  a  matter  of  fact, 
they  are  bordered  with  three  very  distinct  colored  fringes 
when  one  employs  white  light,  and  with  a  still  greater  number 
of  bright  and-  dark  bands  when  one  uses  light  which  is  practi- 
cally homogeneous.  These  fringes  we  shall  call  exterior,  and 
those  which  are  observed  in  the  midst  of  very  narrow  shadows 
we  shall  call  interior  fringes. 

If  one  adopts  the  Newtonian  theory,  he  is  tempted  at  first  to 
explain  the  exterior  fringes  as  produced  by  a  force  which  is  al- 
ternately attractive  and  repulsive,  and  which  has  its  source  in 
the  surface  of  the  body  producing  the  shadows.  I  shall  now 
consider  the  consequences  of  this  theory  and  show  that  its  re- 
sults are  not  justified  by  experiment;  but,  first  of  all,  I  must 
explain  the  experimental  method  which  I  have  employed. 

12.  We  know  that  the  effect  of  a  magnifying-glass  placed  in 
front  of  the  eye  is  to  reproduce  accurately  upon  the  retina  any 
object  or  image  which  is  located  at  its  conjugate  focus  ;  at  least 
F  81 


MEMOIRS    ON 

this  is  the  case  whenever  all  the  rays  which  go  to  make  up  the 
image  are  incident  upon  the  surface  of  the  glass.  In  place, 
then,  of  projecting  the  fringes  upon  a  white  card  or  a  ground 
glass,  one  may  observe  them  directly  with  a  magnifying- 
glass,  and  he  then  sees  them  as  they  are  at  its  focus.  One 
has  then  only  to  look  towards  the  luminous  point  and  place 
the  glass  between  his  eye  and  the  opaque  body  in  such  a  way 
that  the  point  where  the  refracted  rays  cross  each  other  falls 
in  the  middle  of  the  pupil  ;  this  position  is  recognized  by  the 
fact  that  the  entire  surface  of  the  magnifying-glass  appears  to 
be  filled  with  light.  This  method  is  much  preferable  to  the 
other  two  in  that  it  enables  us  to  study  conveniently  phenom- 
ena of  diffraction  in  a  weak  light,  and  has,  at  the  same  time, 
the  further  advantage  of  allowing  us  to  follow  the  exterior 
fringes  right  up  to  their  source.  Using  a  lens  of  2  mm.  focus 
and  light  which  is  practically  homogeneous,  I  have  been  able 
to  follow  these  fringes  very  close  to  their  origin  and  yet  ob- 
serve the  dark  band  of  the  fifth  order.  The  interval  which  sep- 
arates this  band  from  the  edge  of  the  shadow  I  have  measured 
on  the  micrometer  and  find  it  to  be  less  than  0.015  mm.,  while 
the  first  three  fringes  are  comprised  within  a  space  not  exceed- 
0.01  mm.  ;  by  using  a  lens  of  shorter  focus,  one  would  doubt- 
less still  further  diminish  this  distance.  We  may  thus  regard 
the  dark  and  bright  bands  as  beginning  at  the  very  edge  of  the 
opaque  body,  so  long  as  we  do  not  push  the  accuracy  of  our 
measures  beyond  the  hundredth  part  of  a  millimeter — an  ac- 
curacy which  proves  to  be  sufficient,  and  which  it  is  difficult  to 
exceed  except  when  the  fringes  are  somewhat  larger,  as  is  the 
case  with  those  most  frequently  observed. 

13.  This  point  established,  suppose  that  we  measure  the  ex- 
terior fringes  at  any  given  distance  from  the  [opaque]  screen 
and  then  allow  the  luminous  point  to  approach  ;  the  fringes 
are  observed  to  grow  much  larger.  Meanwhile  the  angle  which 
the  incident  ray  passing  through  the  origin  of  the  fringes  makes 
with  the  tangent  drawn  from  the  luminous  point  to  the  edge  of 
the  screen  will  be  almost  zero.  And  since  these  fringes  take 
their  rise  at  a  distance  less  than  0.01  mm.  from  the  edge,  the 
variation  of  this  angle  would  not  be  able  to  sensibly  affect  the 
size  of  the  fringes.  To  explain  this  enlargement,  we  must  there- 
fore assume  that  the  repulsive  force  increases  in  proportion  as 
the  opaque  body  approaches  the  luminous  point.  But  this  is 

82 


THE    WAVE-THEORY    OF    LIGHT 

impossible,  for  the  intensity  of  this  force  can  evidently  depend 
only  upon  the  distance  at  which  the  light  corpuscle  passes  the 
opaque  body,  upon  the  size  and  form  of  the  surface  of  this 
body,  upon  its  density,  mass,  or  nature  ;  and  by  hypothesis 
these  all  remain  constant. 

But  even  if  we  suppose  the  origin  of  the  dark  and  bright 
bands  to  lie  at  a  greater  distance  from  the  edge  of  the  screen, 
a  supposition  which  would  explain  the  fact  that  they  grow 
larger  in  proportion  as  one  approaches  the  luminous  point,  it 
is  still  impossible  to  make  the  results  of  experiment  ag«ree  with 
the  formula  deduced  from  the  [Newtonian]  hypothesis  which 
we  are  here  discussing. 

14.  The  following  table  gives  the  distance  between  the  dark- 
est point  in  the  dark  band  of  the  fourth  order  and  the  edge  of 
the  geometrical  shadow*  for  different  distances  of  the  opaque 
body  from  the  luminous  point.  These  measures  have  been 
taken  with  a  micrometer  eye-piece  which  carries  in  its  focal 
plane  a  silk  fibre,  the  whoje  being  moved  by  a  micrometer 
screw.  By  the  aid  of  a  head  divided  into  one  hundred  parts, 
passing  an  index,  fixed  with  reference  to  the  screw,  one  is 
able  to  read  the  displacement  of  the  silk  thread  to  within 
about  0.01  mm. 


Distance    from    edge  of 

No.  of 
Observation 

Distance  of  luminous  point 
from  opaque  screen 

Distance  of  opaque  body 
from  micrometer 

geometrical  shadow  to 
the  middle  of  the  dark 

band  of  the  fourth  order 

m. 

m. 

mm. 

1 

0.100 

0.7985 

5.96 

2 

0.510 

1.005 

3.84 

3 

1.011 

0.996 

3.12 

4 

2.008 

0.999 

2.71 

5 

3.018 

1.003 

2.56 

6 

4.507 

1.018 

2.49 

7 

6.007 

0.999 

2.40 

These  experiments  were  made  with  practically  homogeneous 
red  light,  which  was  obtained  by  means  of  a  colored  glass  trans- 


*  I  define  geometrical  shadow  as  the  space  included  between  the  straight 
lines  drawn  through  the  luminous  point  and  tangent  to  the  edges  of  the 
screen  ;  this  is  the  shadow  which  the  light  would  project  if  it  were  not 
diffracted. 

83 


MEMOIRS    ON 

mitting  only  the  red  rays  and  a  small  portion  of  the  orange  rays. 
One  might  obtain  more  homogeneous  light  by  use  of  a  prism, 
but  he  would  not  be  so  certain  as  to  its  identity  in  the  various 
observations  —  a  condition  which  it  is  very  necessary  to  sat- 
isfy. 

15.  Let  us  represent  by  a  and  b  the  respective  distances  of 
the  opaque  body  from  the  luminous  point  and  from  the  mi- 
crometer ;  let  d  be  the  distance  from  the  edge  of  the  body  to 
the  origin  of  the  dark  band  of  the  fourth  order,  and  r  the  tan- 
gent of  the  small  angle  of  inflection  resulting  from  the  action 
of  the  repulsive  forces.  We  then  have  the  following  expression* 
for  the  distance  between  the  edge  of  the  geometrical  shadow 
and  the  darkest  point  in  the  dark  band  : 


Now  since  r  and  d  remain  constant  whatever  be  the  distances 
of  the  luminous  point  from  the  opaque  body  and  from  the  mi- 
crometer respectively,  two  observations  suffice  to  determine 
their  value.  Combining  the  first  and  the  last  observations,  we 
find  fl?  =  0.5019  mm.  and  r  =  1.8164.  We  are  thus  compelled  to 
suppose  that  at  its  origin  the  dark  band  of  the  fourth  order  is 
distant  one  -half  a  millimeter  from  the  edge  of  the  opaque 
body.  If,  now,  we  substitute  these  values  in  the  formula  and 
apply  it  to  the  intermediate  observations,  we  obtain  the  follow- 
ing values,  several  of  which  evidently  differ  widely  from  the 
results  of  experiment. 

*  [In  diagram  S  is  luminous  point,  O  is  edge  of  opaque  body,  A  is  edge  of 
geometrical  shadow,  O'  is  origin  of  dark  band  of  fourth  (or  any)  order.    Hence 

Ar>  .   d(a-\-b) 
AB  is  —  --  -  ;  and 
a 

the  centre  of  the  dark 
band  is  a  distance 
br  farther,  where 
A.C  =  br.  The  unit 
which  Fresnel  here 

employs  for  r  is  evidently  one  hundred  times  smaller  than  that  in  which  we 

ordinarily  express  natural  tangents.'] 

84 


THE    WAVE-THEORY    OF 


No.  of 
Observation 

Distance    of     lu- 
minous     point 
from       opaque 
body 

Distance  of  opnque 
body     from     mi- 
crometer 

Distance     between     the 
edge     of    geometrical 
shadow    and    darkest 
point  of  fourth  baud 

Differences 

Observed 

Computed 
from  formula 
d(a+b) 
br+  —  JT~ 

1 

2 
3 

4 
5 
6 

7 

m. 

0.1000 
0.510 
1.011 
2.008 
3.018 
4.507 
6.007 

m. 
0.7985 
1.005 
0.996 
0.999 
1.003 
1.018 
0.999 

m. 
596 
3.84 
3.12 
2.71 
2.56 
249 
2.40 

mm- 

3.32 
281 
2.57 
2.49 
246 

mm. 

-0.52 
-0.31 
-0.14 
-0.07 
-0.03 

16.  In  attributing  the  production  of  fringes  to  the  alternate 
expansion  and  contraction  of  rays  which  pass  very  near  the 
opaque  body,  we  are  led  to  still  another  inference  which  is 
contradicted  by  experiment — viz.,  that  the  centres  of  the  dark 
and  bright  bands  ought  to  lie  along  straight  lines  which  would 
be  the  axes  of  the  expanded  or  condensed  pencils  of  rays. 
But  experiment  shows  that  in  the  case  of  exterior  fringes  their 
trajectories  are  hyperbolas,  of  which  the  curvature  is  quite 
sensible  whenever  the  body  which  produces  the  shadow  is  suf- 
ficiently distant  from  the  luminous  point. 

The  screen  being  placed  at  a  distance  of  3.018  m.  from  the 
luminous  point,  I  measured  in  succession  the  deviation  of  the 
darkest  point  of  the  dark  band  of  the  third  order,  first  at 
0.0017  m.  from  the  screen,  then  at  1.003  m.,  and  lastly  at 
3.995  m.  from  the  screen  ;  and  I  found  for  its  distance  from 
the  edge  of  the  geometrical  shadow  first  0.08  mm.,  secondly  2.20 
mm.,  and  thirdly  5.83  mm.  If,  now,  we  join  the  two  extreme 
points  by  a  straight  line,  we  find  for  the  ordinate  correspond- 
ing to  the  intermediate  point  1.52  mm.  in  place  of  2.20  mm., 
the  difference  being  0.68  mm. — that  is  to  say,  about  one  and 
one-half  times  the  interval  between  the  middle  of  the  third 
and  the  middle  of  the  second  bands.  For  this  interval  at  a 
distance  of  1.003  m.  from  the  opaque  body  was  only  0.42  mm., 
from  which  it  is  evident  that  the  difference  of  0.68  mm.  cannot 
be  attributed  to  an  error  resulting  from  lack  of  definition 
in  the  fringes  observed.  Nor  is  one  able  to  explain  this  dis- 
crepancy by  supposing  an  error  in  the  observation  made  at 
3.995  m.  from  the  opaque  body.  From  the  fact  that  the 

85 


MEMOIRS    ON 

fringes  are  larger,  the  measures  should  be  less  accurate ;  but  in 
repeating  them  several  times  I  find  variations  which  at  most 
amount  to  three  or  four  hundredths  of  a  millimeter.  Even 
supposing  that  there  were  an  error  of  one-half  a  millimeter  in 
this  measure,  it  would  produce  only  a  difference  of  0.13  mm. 
at  a  distance  of  1.003  m.  ;  so  that  experiment  shows  conclu- 
sively that  the  exterior  fringes  lie  on  curved  lines  with  their 
convex  side  outwards. 

The  following  table  gives  these  trajectories,  referred  to  their 
chords,  for  different  series  of  observations,  in  each  of  which 
the  distance  of  the  opaque  body  from  the  luminous  point  re- 
mains constant.  In  the  fourth  series  I  suppose  first  that  the 
chord  joins  the  two  extreme  readings,  and  next  I  suppose  it  to 
be  drawn  from  the  edge  of  the  opaque  body  itself  where  the 
deviation  of  the  fringes  from  their  origin  is,  as  we  have  already 
seen,  very  small.  In  the  other  series  the  chord  joins  the  edge 
of  the  opaque  body  and  the  point  most  distant  from  it. 


Lance  from  lu- 
nous  point  to 
aque  screen, 
the  value  of  a 

gl!s 

<*"  -o  5  o 

Ji* 
llli 

3=.2« 

Ordinates  of  Trajectories  of  dark  bands  referred 
to  their  chords 

saoS 

g&ss 

1st  order 

2d  order 

3d  order 

4th  order 

5th  order 

IST  SERIES 

f  ° 

0 

0 

0 

0 

0 

m. 

mm. 

mm. 

mm. 

mm. 

mm. 

m. 

0.510 

J   0.110 
1   0.501 

019 
0.14 

0.29 
0.21 

035 
0.25 

0.40 
0.30 

0.44 
0.34 

[  1.005 

0 

0 

0 

0 

0 

2o  SERIES 

f   ° 

0 

0 

0 

0 

0 

m. 

mm. 

mm. 

mm. 

mm. 

mm. 

m. 

1    0.116 

0.23 

0.35 

0.42 

0.49 

055 

1.011 

1   0.502 

0.27 

0.40 

0.51 

0.57 

0.63 

0.996 

0.21 

0.30 

0.38 

0.42 

0.49 

[  2.010 

0 

0 

0 

0 

0 

3D  SERIES 

f    o 

0 

0 

0 

0 

0 

m. 

mm. 

mm. 

mm. 

mm. 

mm. 

m. 

2.008 

J   0.118 
1    0.999 

0.26 
0.34 

0.38 
0.48 

0.47 
0.60 

0.54 
0.68 

0.60 
0.76 

[  2.998 

0 

0 

0 

0 

0 

THE    WAVE-THEORY    OF    LIGHT 


4-TH  SERIES  referred  to  the  chord  joining  the  extreme  readings 

m.         r" 

0.0017 

0                0 

0 

0 

0 

mm.              mm. 

linn. 

0.253 

0.30           0.45 

056 





m. 

3.018 

< 

0.500 
1.003 

0.38           0.53 
0.38           0.56 

0.65 

0.68 

z 

— 

1.998 

0.31           0.45 

0.54 

— 

— 

3.002 

0.17           0.23 

0.28 

— 

— 

1  3.995 

0                0 

0 

0 

0 

4TH  SERIES  referred  to  the  chord  drawn  from  the  edge  of  the  opaque  body 

(     0 

0 

0 

0              0 

0 

m. 

mm. 

mm. 

mm. 

00017 

0.04 

0.06 

0.08 

— 

mm. 

mm. 

m 

0.253 

0.34 

0.50 

0.63         0.73 

0.83 

3018 

- 

0.500 

0.41 

0.58 

0  72         0.85 

0.94 

1.003 

041 

0.60 

0.74         0.87 

0.97 

1.998 

0.32 

0.48 

0  57         0  67 

075 

3.002 

0.18 

0.25 

0  30         0  38 

0.39 

13995                0                 0 

0             0 

0 

STH  SERIES 

r     0 

0 

0 

0 

0 

0 

m. 

mm. 

mm. 

mm. 

mm. 

mm. 

4.507 

J  0.131 

1.018 

0.27 
032 

0.40 
0.48 

0.50 
0.59 

0.58 
071 

0.66 
081 

^2.506 

0 

0 

0 

0 

0 

GTH  SERIES 

f    0 

0 

0 

0 

0 

0 

m. 

m. 

mm 

mm. 

mm. 

mm. 

mm. 

6007 

1  0.117 

023 

033 

0.42 

0.49 

0.53 

LO.  999 

0 

0 

0 

0 

0 

It  is  thus  evident  that  the  hypothesis  of  contraction  and 
expansion  produced  by  the  action  of  the  body  upon  rays  of 
light  is  insufficient  to  explain  the  phenomena  of  diffraction. 
Introducing  the  principle  of  interference,  however,  we  are  able 
to  predict  not  only  the  variation  in  size  of  the  exterior  fringes 
when  the  screen  is  made  to  approach  or  recede  from  the  lumi- 
nous point,  but  also  the  curved  path  of  the  bright  and  dark 
bands.  The  law  of  interference,  or  the  mutual  influence  of 
rays  of  light,  is  an  immediate  consequence  of  the  wave-theory; 
not  only  so,  but  it  is  proved  or  confirmed  by  so  many  different 
experiments  that  it  is  really  one  of  the  best-established  prin- 
ciples of  optics. 

17.   Grimaldi  was  the  first  to  observe  the  effect  which  rays 

87 


MEMOIRS    ON 

of  light  produce  upon  one  another.  Recently  the  distinguish- 
ed Dr.  Thomas  Young  has  shown  by  a  simple  and  ingenious 
experiment  that  the  interior  fringes  are  produced  by  the  meet- 
ing of  rays  inflected  at  each  side  of  the  opaque  body.  This 
he  proved  by  using  a  screen  to  intercept  one  of  the  two  pen- 
cils of  light;  and  in  this  way  lie  was  able  to  make  the  interior 
fringes  completely  vanish,  whatever  might  be  the  form,  mass, 
or  nature  of  the  screen,  and  whether  he  intercepted  the  lu- 
minous pencil  before  or  after  its  passage  into  the  shadow. 

18.  Brighter  and  sharper  fringes  may  be  produced  by  cut- 
ting two  parallel  slits  close  together  in  a  piece  of  cardboard 
or  a  sheet  of  metal,  and  placing  the  screen  thus  prepared  in 
front  of  the  luminous  point.     We  may  then  observe,  by  use 
of  a  magnifying -glass  between  the  opaque  body  and  the  eye, 
that  the  shadow  is  filled  with  a  large  number  of  very  sharp- 
colored  fringes  so  long  as  the  light  shines  through  both  open- 
ings at  the  same  time,  but  these  disappear  whenever  the  light 
is  cut  off  from  one  of  the  slits. 

19.  If  we  allow  two  pencils  of  light,  each  coming  from  the 
same  source  and  regularly  reflected  by  two  metallic  mirrors, 
to  meet  under  a  very  small  angle,  we  obtain  similar  fringes, 
the  colors  of  which  are  even  purer  and  more  brilliant  than 
before.     To  obtain  these  bands,  it  is  necessary  to  be  very  care- 
ful that  in  the  region  where  the  two  mirrors  come  into  con- 
tact, or  at  least  throughout  a  portion  of  their  line  of  contact,, 
the  surface  of  the  one  is  not  shifted  sensibly  past  that  of  the 
other.     This  is  necessary  in  order  that  the  difference  of  path 
traversed  by  two  reflected  rays  meeting  in  the  area  common 
to  the  two  luminous*  fields  may  be  very  small.     I  may  re- 
mark in  passing  that  the   theory  of   interference  alone  will 

*  In  the  case  of  white  light,  or  even  in  light  as  homogeneous  as  possible, 
the  number  of  fringes  which  one  can  see  is  always  limited,  because  even 
when  the  light  has  reached  a  degree  of  simplicity  as  great  as  possible 
without  too  far  diminishing  its  intensity,  it  is  still  composed  of  rays  which 
are  heterogeneous;  and  since  the  bright  and  dark  bands  thus  produced  do 
not  all  have  the  same  size,  they  encroach  the  one  upon  the  other  in  pro- 
portion as  their  order  increases,  and  finally  they  completely  destroy  each 
other  ;  and  this  is  why  one  does  not  see  any  fringes  when  the  difference 
of  paths  becomes  slightly  sensible.  Concerning  the  details  of  this  ex- 
periment and  its  explanation  on  the  principle  of  interference,  see  the  arti- 
cle upon  Light  in  the  French  translation  of  Thomson's  Chemistry,  already 
cited. 


THE    WAVE-THEORY    OF    LIGHT 

explain  this  experiment,  and  that  the  experiment  calls  for 
manipulation  so  delicate  and  effort  so  continued  that  it  is 
almost  impossible  that  one  should  strike  upon  it  by  accident. 

If  we  raise  one  of  the  mirrors  or  intercept  the  light  which  it 
reflects  either  before  or  after  reflection,  the  fringes  disappear 
as  in  the  preceding  case.  This  furnishes  still  further  evidence 
that  the  fringes  are  produced,  not  by  the  action  of  the  edges 
of  the  mirrors,  but  by  the  meeting  of  two  pencils  of  light. 
For  these  fringes  are  always  at  right  angles  to  the  line  which 
joins  the  two  images  of  the  luminous  point,  whatever  be  its 
inclination  with  respect  to  these  edges,  at  least  throughout  the 
extent  of  the  area  which  is  common  to  the  two  regularly  re- 
flected pencils.* 

20.  Since  the  fringes  which  one  sees  in  the  interior  of 
the  shadow  of  a  very  narrow  body  and  those  which  one  ob- 
tains by  the  use  of  two  mirrors  result  evidently  from  the  mut- 
ual influence  of  rays  of  light,  analogy  would  indicate  that  the 
same  thing  ought  to  be  true  for  the  exterior  fringes  of  the 
shadows  of  bodies  illuminated  by  a  point  source.  The  first 
explanation  which  occurs  to  one  is  that  these  fringes  are  pro- 
duced by  the  interference  of  direct  rays  with  those  which  are 
reflected  at  the  edge  of  the  opaque  body,  while  the  interior 
fringes  result  from  the  combined  action  of  rays  inflected  into 
the  shadow  from  the  two  sides  of  the  opaque  body,  these  in- 
flected rays  having  their  origin  either  at  the  surface  or  at 
points  indefinitely  near  it.  This  appears  to  be  the  opinion  of 
Mr.  Young,  and  it  was  at  first  my  o\vn  opinion  ;  but  a  closer 
examination  of  the  phenomena  convinced  me  of  its  falsity. 
Nevertheless,  I  propose  to  follow  it  to  its  logical  conclusion 
and  to  state  the  formula  which  I  have  derived  in  order  to  facil- 
itate comparison  of  this  theory  with  that  which  I  offer  as  a 
substitute. 

Let  R,  Fig.  14,  be  the  radiant  point,  AA'  the  opaque  body,  and 
FT'  either  a  white  screen  upon  which  the  shadow  of  this  body 
falls  or  the  focal  plane  of  a  magnifying-glass  with  which  the 

*  When  the  fringes  extend  outside,  all  their  exterior  portions  resulting 
from  the  meeting  of  rays  regularly  reflected  by  one  of  the  mirrors  and  rays 
inflected  near  the  edge  of  the  other  should  have  different  directions.  If 
one  observes  this  phenomenon  carefully,  he  will  see  that  the  form  and 
position  of  the  fringes  are  in  each  case  in  accord  with  the  theory  of  inter- 
ference. 


MEMOIRS    ON 


fringes  are  observed.  RT  and  RT'  are  rays  tangent  at  the 
edge  of  the  opaque  body,  T  and  T'  being  the  limits  of  the  ge- 
ometrical shadow.  Let  us  indicate  by 
a  the  distance  RB  from  the  luminous 
point  to  the  opaque  body,  by  b  the  dis- 
tance BO  of  the  body  from  the  white 
screen,  and  by  c  its  diameter,  AA', 
which  we  shall  consider  very  small  com- 
pared with  the  distances  a  and  b.  This 
assumption  is  made  in  order  that  we 
may  measure  the  size  of  the  fringes 
either  in  a  plane  perpendicular  to  RT 
or  perpendicular  to  the  line  RC, 
which  passes  through  the  middle  of  the 
shadow. 

With  these  conventions  we  shall  con- 
sider, first,  the  exterior  fringes.  Let  F  be  any  point  on  the  re- 
ceiving screen  outside  the  shadow.  The  difference  of  path 
traversed  by  the  direct  ray,  and  by  the  ray  reflected  at  the  edge 
of  the  opaque  body,  and  meeting  the  direct  ray  at  this  point, 
is  RA  +  AF  — RF.  Let  us  represent  FT  by  x,  and  express  in 
series  the  values  of  RF,  AR,  and  AF.  Then,  if  we  neglect  all 
terms  involving  any  power  of  x  or  of  c  higher  than  the  second, 
since  they  are  very  small  compared  with  distances  a  and  b,  the 
terms  which  contain  c  will  disappear  and  we  shall  have  for  the 
difference  of  path  traversed 


*=-< 


whence  follows 


21.  If  we  call  X  the  length  of  a  light-wave,  that  is  to  say, 
the  distance  between  two  points  in  the  ether  where  vibrations 
of  the  same  kind  are  occurring  at  the  same  time  and  in  the 
same  sense,  then  A/2  will  be  the  distance  between  two  ether 
particles  whose  velocities  of  vibration  are  at  any  one  instant 
equal  but  oppositely  directed.  Thus  two  trains  of  waves  sepa- 
rated by  an  interval  equal  to  X  are  in  perfect  accord  as  to  their 
vibrations;  but  when  the  distance  between  corresponding  points 
is  A/2,  then  their  vibrations  are  directly  opposed.  Accordingly 

90 


THE    WAVE-THEORY    OF    LIGHT 

the  above  formula  gives  for  the  value  of  x,  corresponding  to 
the  centre  of  the  dark  band  of  the  first  order,  the  following 

value  :  \J — * '- ;  while  observation  shows  that,  as  a  matter 

v  ct 

of  fact,  this  is  the  brightest  part  of  the  first  fringe.  On  the 
same  theory,  the  edge  of  the  geometrical  shadow,  where  the 
difference  of  path  vanishes,  ought  to  be  brighter  than  the  rest 
of  the  fringe,  while,  as  a  matter  of  fact,  this  is  precisely  the 
darkest  region  outside  the  geometrical  shadow.  In  general, 
the  position  of  the  dark  and  bright  bands  deduced  from  this 
formula  is  almost  exactly  the  inverse  of  that  determined  by 
experiment.  This  is  the  first  difficulty  presented  by  this 
theory.  To  avoid  it,  we  must  suppose  that  the  rays  reflected 
at  the  edge  of  the  screen  suffer  the  loss  of  half  a  wave-length; 
adding  A/2  to  the  difference  of  path,  d,  the  general  expression 
becomes 

y  ~      a 

Replacing  d  in  this  formula  by  A/2,  3  A/2,  5  A/2,  7  A/2,  etc.,  we 
have  for  the  values  of  x  corresponding  to  dark  bands  of  the 
first,  second,  third,  fourth,  etc.,  orders: 


/2\b(a  +  d)       /±\b(a  +  b)       /Q\b(a+b)     . 

v  -  —^r     v    —  <r~     V~  —  ^T->  V~  —^-->  etc- 

These  formulae  appear  to  agree  fairly  well  with  the  observations; 
however,  closer  measurements  show  that  the  ratios  between  the 
sizes  of  the  fringes  derived  from  these  expressions  are  not  ex- 
actly correct,  as  we  shall  see  later. 

22.  I  pass  now  to  the  consideration  of  interior  fringes  pro- 
duced in  the  shadow  by  the  meeting  of  two  pencils  of  light 
inflected  at  A  and  A'.  Let  M,  Fig.  14,  be  any  point  located  in 
the  interior  of  the  shadow  ;  the  intensity  of  the  light  at  this 
point  depends  upon  the  amount  of  disagreement  between  the 
vibrations  of  the  rays  AM  and  A'M,  which  meet  at  this  point, 
or  upon  the  difference  of  path  A'M  —  AM.  I  shall  denote  by  x 
the  distance  MC  of  the  point  M  from  the  middle  of  the  shadow, 
and  by  d  the  difference  of  paths,  and  hence 


Expanding  the  radicals  and  neglecting  the  higher  powers  of 
x,  since  this  quantity  is  very  small  compared  with  b,  we  have 

d—cxjb, 
91 


MEMOIRS    ON 

or  x  =  bd/c. 

If  in  place  of  d  in  this  expression  we  substitute  successively  X/2, 
3X/2,  5X/2,  7X/2,  etc.,  we  obtain  the  values  of  x  correspond- 
ing to  dark  bands  of  the  first,  second,  third,  fourth,  etc., 
order,  namely, 

b\        3£X         5b\         7#X      , 
7\~>       ~T» — '        ~T» — >       ~z\ — >  etc., 
2c         2c  2c  2c 

and  consequently  for  the  distance  between  the  middle  points  of 
two  consecutive  dark  bands,  b\/c. 

The  general  expression  for  n  such  intervals  is,  therefore, 
rib\lc. 

23.  So  long  as  the  extreme   fringes  are  sufficiently  distant 
from  the  edges  of  the  shadow,  this  formula  agrees  fairly  well 
with  experiment ;   but  when  they  approach  very  near  or  pass 
beyond  the   edges,  one   detects   a   slight    difference   between 
their  actual  position  and  that  deduced  from  the  formula.     In 
general,  the  calculated  values  are  always  a  little  larger  than 
the  observed.     The  reason  for  this  I  shall  show  when  we  come 
to  the  true  theory  of  diffraction.     It  also  follows   from  this 
formula  that  the  size  of  the  interior  fringes  ought  to  be  entire- 
ly independent  of  the  distance,  a,  of  the  luminous  point  from 
the  opaque  body;  this  prediction,  however,  is  not  completely 
verified  by  experiment,  especially  when  the  fringes  completely 
fill  the  shadow;  their  position  then  varies  distinctly  with  the 
distance  a. 

24.  According  to  the  formula 


which  we  have  just  derived  for  the  exterior  fringes,  their 
position  depends  upon  a  as  well  as  upon  b.  Experiment  shows 
that,  in  fact,  their  size  increases  or  diminishes  according  as  the 
opaque  body  approaches  or  recedes  from  the  luminous  point, 
and  that  the  ratios  between  the  different  sizes  of  one  and  the 
same  fringe  deduced  from  the  formula  are  precisely  those  given 
by  observation.  But  the  most  remarkable  inference  from  this 
formula  is  that,  when  a  remains  constant,  the  distance  of  any 
dark  or  bright  band  from  the  edge  of  the  geometrical  shadow 
is  not  directly  proportional  to  b  as  in  the  case  of  interior 
fringes,  but  varies  in  such  a  way  that  this  band  traces  out,  not 
a  straight  line,  but  a  hyperbola  of  sensible  curvature.  This  is 

92 


THE    WAVE-THEORY    OF    LIGHT 

also  confirmed  by  experiment,  as  may  be  seen  from  the  observa- 
tions given  above. 

Considering  the  striking  agreement  of  these  formulae  with 
experiment,  it  is  natural  to  suppose  that  they  are  accurate  ex- 
pressions of  fact,  and  therefore  natural  to  attribute  any  small 
differences  between  calculated  and  observed  values  to  the  errors 
which  are  unavoidable  in  such  delicate  measurements.* 

But  a  closer  examination  of  the  hypotheses  from  which  they 
are  derived,  and  of  the  inferences  derivable  from  them,  shows 
that  they  do  not  agree  with  the  facts  of  nature. 

25.  If  the  fringes  at  the  edge  of  a  shadow  are  really  due  to 
the  meeting  of  the  direct  rays  with  those  reflected  at  the  edge 
of  the  screen,  their  intensity  ought  to  depend  upon  the  area 
and  the  curvature  of  its  surface,  and  the  fringes  produced  by 
the  back  of  a  razor,  for  instance,  ought  to  be  much  more  vis- 

*  It  might  appear  at  first  sight  that  one  would  be  able  to  adapt  this 
theory  to  the  ideas  of  Newton  by  introducing  the  principle  of  interference, 
as  I  have  indicated  above  ;  but  besides  the  complication  of  fundamental 
hypotheses  and  the  small  probability  of  any  of  them,  this  principle,  it  ap- 
pears to  me,  would  lead  to  consequences  which  contradict  the  emission- 
theory. 

M.  Arago  has  remarked  that  the  interposition  of  a  thin  transparent  plate 
at  the  edge  of  an  opaque  body  sufficiently  narrow  to  produce  interior 
fringes  in  its  shadow  displaces  these  fringes  and  shifts  them  towards  the 
side  [see  paper  by  Arago  (Ann.  Chim.  et  Phys.,\.,  p.  199,  1816)]  where  is 
placed  the  transparent  plate.  This  being  so,  it  follows  from  the  principle 
of  interference  that  the  rays  which  have  traversed  the  plate  have  been  re- 
tarded in  their  path,  because  the  same  fringes  in  each  case  must  correspond 
to  equal  intervals  between  the  times  of  arrival  of  rays.  This  inference  at 
once  confirms  the  wave-theory  and  manifestly  contradicts  the  emission- 
theory,  in  which  one  is  compelled  to  assume  that  light  travels  more  rapidly 
in  dense  than  in  rare  media. 

This  objection  can  be  avoided  only  by  substituting  for  difference  of 
path  difference  of  "fit";  but  we  lose  all  that  was  gained  by  the  principle 
of  interference  in  thus  replacing  a  sharp  idea  by  a  hazy  one,  a  satisfactory 
explanation  by  one  which  does  not  aid  our  understanding  of  the  phenome- 
na ;  for  one  can  readily  see  how  two  light  particles  striking  the  retina  at  the 
same  point  may  produce  sensations  more  or  less  intense,  according  as  the 
interval  of  time  which  separates  two  consecutive  impacts  is  sufficient  to 
produce  unison  or  dissonance  between  the  vibrations  at  the  optic  nerve; 
while  it  is  by  no  means  so  easy  to  see  how  this  effect  could  be  produced 
by  a  difference  of  "  fit "  between  two  light  particles,  or  how  by  simultaneous 
impact  on  the  optic  nerve  they  would  produce  no  effect  at  all  when  they 
were  in  opposite  "  fits,"  even  though  their  mechanical  impacts  were  in  per- 
fect unison. 


MEMOIRS    ON 


ible  than  those  produced  by  the  edge;  but,  using  a  magnifying- 
glass  at  a  distance  of  some  centimeters,  one  detects  practi- 
cally no  difference  in  intensity  in  these  two  cases.  This  test 
is  more  easily  made  by  using  a  steel  plate  one  edge  of  which 
is  round  throughout  a  part  of  its  length  and  sharp  through- 
out the  remainder  of  its  length,  these  two  edges  lying  in 
the  same  straight  line.  One  is  thus  easily  convinced  that 
the  fringes  have  the  same  intensity  throughout  their  entire 
length. 

26.  We  know  that  under  large  angles  of  incidence  dull  sur- 
faces reflect  light  almost  as  well  as  polished  mirrors.     This  is 
easily  explained  either  on  the  emission-theory  or  on  the  wave- 
theory.     But  although  one  can  understand  how  difference  of 
polish  cuts  a  small  figure  when  the  angle  of  incidence  is  large, 
it  is  not  easy  to  see  how  the  intensity  of  the  reflected  light  can 
be   independent   of  the    curvature   of   the  reflecting  surface  ; 
indeed,  it  is  clear  that  as  the  radius  of  curvature  diminishes 
the  reflected  rays  will  diverge  more  and   more,  whatever-  be 
their  angle  of  incidence. 

27.  Not  only  so,  but  I   have   convinced  myself  by  another 
simple  experiment  of  the  incorrectness  of  the  hypothesis  which 
I  had  first  adopted,  and  which  I  am  now  opposing.    I  cut  a 
sheet  of  copper  into  the  shape  represented  in  Fig.   15,   and 
placed  it  in  a  dark  room   about  four  meters   in  front  of  a 
luminous  point,  and  examined  its  shadow  with  a  magnifying- 

glass.  What  I  observed,  on  slowly  re- 
ceding, was  as  follows:  When  the  large 
fringes  produced  by  each  of  the  very 
narrow  openings  CEE'C'  and  DFF'D' 
had  spread  out  into  the  geometrical 
shadow  of  CDFE,  which  received  prac- 
tically only  white  light  from  each  sep- 
arate slit,  the  interior  fringes  produced 
by  the  meeting  of  these  two  pencils  of 
light  showed  colors  much  sharper  and 


Fig  i 5. 


purer  than  the  interior  fringes  of  the  shadow  of  ABDO,  and 
were,  at  the  same  time,  much  brighter.  On  receding  still 
farther,  I  noticed  that  the  light  diminished  throughout  the 
whole  of  the  shadow  of  ABFE,  but  much  more  rapidly  back 
of  EFDC  than  in  the  upper  part  of  the  shadow,  so  that  there 
was  one  particular  instant  when  the  intensity  of  the  light  ap- 

94 


THE    WAVE-THEORY    OF    LIGHT 

peared  to  be  the  same  above  and  below,  after  which  the  fringes 
remained  less  intense  in  the  lower*  part,  although  their  colors 
were  always  much  purer. 

If,  now,  the  only  inflected  light  were  that  which  grazed  the 
edges  of  the  opaque  bodies,  the  fringes  of  the  upper  part  ought 
to  be  sharper  and  ought  to  show  purer  colors  than  those  of 
the  lower  part ;  for*  the  first  are  produced  by  the  meeting  of 
two  systems  of  waves  which  have  their  centres  upon  the  edges 
AC  and  BD,  while  the  others  are  formed  by  the  meeting  of 
four  systems  of  waves  having  their  origin  at  the  edges  C'E', 
CE,  DF,  DF';  and  this  would  necessarily  diminish  the  differ- 
ence of  intensity  between  the  dark  and  bright  bands,  in  the 
case  of  homogeneous  light,  or  the  purity  of  the  colors,  in  the 
case  of  white  light,  because  the  fringes  produced  by  the  rays 
reflected  and  inflected  at  C'E'  and  DF  would  not  exactly  coin- 
cide with  those  produced  by  the  meeting  of  rays  coming  from 
CE  and  D'F'.  Now  experiment  shows,  as  I  have  just  said,  that 
exactly  the  reverse  of  this  is  true.  One  might  explain  on  this 
same  hypothesis  how  it  happens  that  the  shadow  of  ECDF  is 
much  brighter  than  that  of  ABDC  arising  from  the  double 
source  of  light  presented  by  the  two  edges  of  each  slit ;  but  from 
this  it  would  follow  that  the  lower  part  ought  always  to  be 
brighter,  and  we  have  just  seen  that  this  is  not  the  fact. 

28.  From  the  experiments  which  I  have  just  described  it  is 
evident  that  we  cannot  attribute  the  phenomena  of  diffraction 
solely  to  rays  which  graze  the  edge  of  the  body  ;  but  we  must, 
on  the  contrary,  admit  that  there  is  an  infinitude  of  other  rays 
sensibly  distant  from  the  body  and  yet  deviated  from  their 
original  direction,  so  as  to  meet  and  form  these  fringes. 

29.  The  spreading  out  of  a  pencil  of  light  in  passing  through 
a  very  narrow  opening  shows  in  an  even  better  manner  that  the 
inflection  of  light  occurs  at  a  sensible  distance  from  the  edges 


*  In  order  that  this  difference  of  intensity  between  the  two  parts  of  the 
shadow  shall  be  as  marked  as  possible,  it  is  necessary  that  the  slits  CE  aud 
DF  be  very  narrow  as  compared  with  the  distance  which  separates  them, 
and  that  the  sheet  of  copper  should  be  as  far  away  as  possible  from  the 
luminous  point. 

[In  repeating  this  experiment,  it  will  be  found  very  convenient  to  use,  in- 
stead of  sheet  copper,  an  unfieed  photographic  plate :  lantern  slide  is  best. 
The  two  slits  can  be  cut  either  with  a  pocket-knife  or,  better  still,  by  means  of  a 
dividing  engine.] 

95 


MEMOIRS    ON 

of  the  diaphragm.  It  was  in  the  consideration  of  this  phenom- 
enon that  I  discovered  the  error  into  which  I  had  previously 
fallen.  When  one  brings  the  edges  of  two  opaque  screens  very 
close  together  in  front  of  a  luminous  point  in  a  dark  room,  he 
observes  that  the  region  illuminated  by  the  aperture  greatly  in- 
creases. Such  screens  were  Newton's  two  knife-edges.  I  shall 
suppose,  as  in  his-  experiment,  that  the  edges  of  the  aperture 
are  thin  and  perfectly  sharp  ;  not  that  this  has  any  effect  upon 
the  phenomena,  but  simply  for  making  clearer  the  conclusion 
which  is  to  be  drawn.  The  small  number  of  rays  which  graze 
these  sharp  edges,  being  spread  out  over  a  rather  large  area, 
could  produce  only  an  insensible  amount  of  illumination,  or,  at 
most,  an  exceedingly  feeble  light,  and  in  the  midst  of  it  one 
ought  to  see  a  bright  band  traced  out  by  the  pencil  of  direct 
rays.  This,  however,  is  not  the  fact ;  for  white  light  of  almost 
uniform  intensity  fills  a  space  much  larger  than  the  projection 
of  the  aperture,*  and  gradually  grows  weaker,  shading  into  the 
dark  bands  of  the  first  order.  It  was  doubtless  in  order  to  ac- 
count for  the  large  amount  of  light  inflected  that  Newton  sup- 
posed the  action  of  the  body  upon  rays  of  light  to  extend  to 
sensible  distances,  but  this  hypothesis  will  not  bear  careful 
scrutiny. 

30.  If  the  expansion  of  a  pencil  of  light  which  passes  through 
a  narrow  opening  were  brought  about  by  attractive  and  repul- 
sive forces  having  their  origin  at  the  edges  of  the  aperture,  the 
intensity  of  these  forces,  and  consequently  their  effect  upon 
the  light,  would  necessarily  vary  with  the  nature,  the  mass,  and 
the  surface  of  the  edges  of  the  screen.  All  forces  produced  by 
a  body  acting  at  a  sensible  distance  and  taking  their  rise  in  any 
considerable  extent  of  its  mass  or  its  surface  would  depend 
upon  the  relative  positions  and  upon  the  number  of  particles 
contained  within  this  sphere  of  activity,  or,  what  is  the  same 
thing,  upon  the  shape  of  the  surface.  If,  then,  the  phenomena 
in  question  are  due  to  the  action  of  such  forces,  one  would  ex- 
pect that,  on  placing  a  sharp  body  opposite  a  round  body,  the 
rays  of  light  would  be  inflected  more  to  the  one  side  than  the 

*  The  illuminated  space  increases  so  rapidly  in  comparison  with  the  width 

.  of  the  conical  projection  of  the  aperture  as  the  receiving  screen  recedes  from 

the  aperture,  and  likewise  when  the  aperture  itself  is  further  withdrawn 

from  the  luminous  point,  that  by  making  these  two  distances  sufficiently 

great  one  can  obtain  the  same  effect  with  an  opening  of  any  size. 

96 


THE    WAVE-THEORY    OF    LIGHT 

other  ;  but,  as  I  have  shown  by  a  very  simple  experiment,  this 
is  not  the  fact.  I  passed  a  pencil  of  rays  between  two  steel 
plates  whose  vertical  edges  were  brought  very  close  together 
and  were  carefully  straightened  throughout  their  entire  length. 
A  part  of  each  edge  was  sharp,  the  rest  of  it  round,  and  these 
edges  were  arranged  so  that  the  round  portion  of  one  plate  cor- 
responded to  the  sharp  one  of  the  other.  Thus,  if  a  sharp  edge 
were  located  on  the  right  in  the  upper  part  of  the  opening,  an- 
other was  located  on  the  left  in  the  lower  part,  so  that  if  there 
had  been  any  difference  in  the  action  of  the  two  edges  upon  the 
rays,  I  should  have  noticed  it  in  the  relative  positions  of  the 
upper  and  lower  parts  of  the  bright  interval  at  the  middle,  and 
especially  in  the  fringes  in  that  neighborhood,  as  they  would  be 
interrupted  at  the  point  of  passage  from  the  sharp  to  the  round 
edge  ;.  but,  on  observing  them  closely,  I  noted  that  they  were 
perfectly  straight  throughout  their  entire  length,  even  at  the 
bright  interval  in  the  middle,  exactly  in  the  same  way  as  when 
two  edges  of  the  same  kind  are  opposed  one  to  the  other.  The 
experiment  may  be  varied  by  using  plates  made  of  two  different 
substances,  but  the  result*  obtained  will  certainly  remain  the 
same. 

31.  All  the  experiments  which  I  have  tried  so  far  have 
shown  that  the  nature  of  the  body  interposed  has  in  other 
respects  no  more  influence  upon  the  inflection  of  light  than  is 
exerted  by  the  mass  or  the  shape  of  the  two  edges.  I  shall 
cite  only  one  experiment,  in  which  I  have  taken  every  precau- 
tion necessary  to  determine  the  correctness  of  this  principle, 
which,  indeed,  is  already  well  established  by  the  preceding 
experiment. 

I  covered  an  unsilvered  mirror  with  a  layer  of  India  ink 
spread  over  a  thin  layer  of  paper,  forming  together  a  thickness 
of  one-tenth  of  a  millimeter.  With  a  sharp  point  I  traced  two 
parallel  lines,  and  then  carefully  removed  from  between  these 
two  lines  the  paper  and  the  India  ink  which  adhered  to  the 
surface  of  the  glass.  This  aperture,  as  measured  by  the  microm- 

*  Messrs.  Berthollet  and  Mains  found  a  long  while  ago  that  the  nature 
of  the  body  had  no  effect  upon  the  diffraction  of  light.  For  screens  they 
employed  plates  composed  of  different  substances,  with  edges  made  up, 
for  instance,  of  very  dense  metal  and  a  piece  of  ivory  ;  but  they  had  no 
means  of  observation  so  convenient  and  accurate  iis  mine,  and  consequently 
one  might  suspect  that  some  small  difference  might  have  escaped  them. 
G  97 


MEMOIRS    ON 

eter,  was  1.17  mm.  I  then  placed  opposite  each  other  two 
copper  cylinders,  each  having  a  diameter  of  14.5  mm.,  and  by 
means  of  a  graduated  wedge  I  made  the  interval  between  these 
cylinders  also  1.1?  mm.  The  cylinders,  placed  alongside  the 
blackened  glass,  were  at  a  distance  of  4.015  m.  from  the  lumi- 
nous point,  and  at  a  distance  of  1.663  m.  from  the  micrometer. 
I  then  measured  the  size  of  the  fringes  produced  by  these  two 
openings,  and  found  that  they  were  absolutely  the  same.  The 
following  are  the  results  of  the  two  observations  made  with 
white  light : 

The  distance  between  the  darkest  point  ~\  mm. 

of  the  two  dark  bands  of  the  first  I  First  reading,     1.49 

order  at  the  point  of  separation  of  [Second  reading,  1.49 

the  brownish  red  from  the  violet    .  J 
The  interval  between  the  two  fringes  1  Firgt  reading>     3.33 

of  the  second  order  at  the  point  of  Y  Second  reading)  3>22 

separation  of  red  and  green  .   ' 

It  is  hardly  possible  that  two  sets  of  circumstances  should 
differ  more  than  these  as  regards  the  mass  and  the  nature  of 
the  edges  of  the  aperture.  In  the  one  case  there  is  a  single 
layer  of  India  ink  producing  the  fringes,  for  the  glass  to  which 
it  adheres  completely  fills  the  aperture  ;  in  the  other  case  we 
have  two  massive  cylinders  of  copper,  14.5  mm.  in  diameter, 
giving  us  an  aperture  whose  edges  have  very  considerable 
masses  and  areas,  but  we  observe  110  difference  in  the  expan- 
sion of  the  pencil  of  light. 

32.  It  is  therefore  certain  that  the  phenomena  of  diffraction 
do  not  at  all  depend  upon  the  nature,  the  mass,  or  the  shape  of 
the  body  which  intercepts  the  light,*  but  only  upon  the  size  of 
the  intercepting  body  or  upon  the  size  of  the  aperture  through 
which  it  passes.  We  must,  therefore,  reject  any  hypothesis 
which  assigns  these  phenomena  to  attractive  and  repulsive 

*  This  is  so,  at  least  provided  one  does  not  consider  the  shadow  too  close 
up  to  the  edge  of  the  screen,  or  provided  the  surface  grazed  by  the  rays  of 
light  has  not  too  large  an  area  compared  with  this  distance  ;  for  in  this 
case  it  may  happen  that  the  reflected  rays  sensibly  affect  the  phenomenon 
— as,  for  instance,  occurs  when  the  surface  grazed  by  the  rays  is  a  plane 
mirror  of  one  or  two  decimeters  in  size  and  when  one  observes  the  fringes 
at  a  short  distance.  Besides,  there  would  then  be  successive  diffractions 
over  an  area  too  considerable  for  one  to  neglect. 


THE    WAVE-THEORY    OF    LIGHT 

forces  whose  action  extends  to  a  distance  from  the  body  as 
great  as  that  at  which  rays  are  inflected.  We  are  equally 
unable  to  admit  that  diffraction  is  caused  by  a  shallow  atmos- 
phere which  has  the  same  thickness  as  the  sphere  of  activity 
of  these  forces,  and  whose  refractive  index  differs  from  that  of 
the  neighboring  medium  ;  for  this  second  hypothesis,  like  the 
first,  would  lead  us.'to  think  that  the  inflection  of  light  ought 
to  vary  with  the  form  and  the  nature  of  the  edge  of  the  screen, 
and  ought  not  to  be  the  same,  for  instance,  at  the  edge  and  at 
the  back  of  a  razor.  Now,  on  the  emission-theory  it  is  impos- 
sible to  explain  in  any  other  manner  the  expansion  of  a  beam  of 
light  passing  through  a  narrow  opening,  and  this  expansion  is 
a  well-established  fact.*  Consequently,  the  phenomena  of  dif- 
fraction cannot  be  explained  on  the  emission-theory. 

SECTION  II 

33.  In  the  first  section  of  this  memoir  I  have  shown  that 
the  corpuscular  theory,  and  even  the  principle  of  interference 
when  applied  only  to  direct  rays  and  to  rays  reflected  or  in- 
flected at  the  very  edge  of  the  opaque  screen,  is  incompetent 
to  explain  the  phenomena  of  diffraction.  I  now  propose  to 
show  that  we  may  find  a  satisfactory  explanation  and  a  general 
theory  in  terms  of  waves,  without  recourse  to  any  auxiliary 
hypothesis,  by  basing  everything  upon  the  principle  of  Huygens 
and  upon  that  of  interference,  both  of  which  are  inferences 
from  the  fundamental  hypothesis. 

Admitting  that  light  consists  in  vibrations  of  the  ether  sim- 
ilar to  sound-waves,  we  can  easily  account  for  the  inflection 
of  rays  of  light  at  sensible  distances  from  the  diffracting 
body.  For  when  any  small  portion  of  an  elastic  fluid  under- 

*  The  rise  of  a  liquid  in  a  capillary  tube  occurs  between  two  surfaces 
separated  by  a  finite  distance,  although  the  attraction  which  these  sur- 
faces exert  upon  the  liquid  extends  only  to  an  infinitely  small  distance. 
The  reason  of  this  is,  that  the  molecules  of  the  liquid,  attracted  by  the 
surface  of  the  tube,  also  in  their  turn  attract  other  molecules  of  the  liquid 
situated  within  their  sphere  of  action,  and  so  on,  step  by  step  ;  but  in  the 
emission-theory  an  analogous  explanation  is  not  admissible,  for  the  funda- 
mental hypothesis  is  that  the  luminous  particles  never  exert  any  sensible 
effect  upon  the  path  of  neighboring  particles.  No  interdependence  of 
motion  is  here  admissible,  for  such  an  assumption  would  be  the  assumption 
of  a  fluid  medium. 

99  s^0^* 

S  V*  OJ-   THB         -Y 

I  UNIVERSITY 

\£f 


MEMOIRS    ON 

goes  condensation,  for  instance,  it  tends  to  expand  in  all  direc- 
tions ;  and  if  throughout  the  entire  wave  the  particles  are  dis- 
placed only  along  the  normal,  the  result  would  be  that  all 
points  of  the  wave  lying  upon  the  same  spherical  surface  would 
simultaneously  suffer  the  same  condensation  or  expansion,  thus 
leaving  the  transverse  pressures  in  equilibrium  ;  but  when  a 
portion  of  the  wave-front  is  intercepted  or  retarded  in  its  path 
by  interposing  an  opaque  or  transparent  screen,  it  is  easily  seen 
that  this  transverse  equilibrium  is  destroyed  and  that  various 
points  of  the  wave  may  now  send  out  rays  along  new  direc- 
tions. 

To  follow  by  analytical  mechanics  all  the  various  changes 
which  a  wave-front  undergoes  from  the  instant  at  which  a  part 
of  it  is  intercepted  by  a  screen  would  be  an  exceedingly  diffi- 
cult task,  and  we  do  not  propose  to  derive  the  laws  of  diffrac- 
tion in  this  manner,  nor  do  we  propose  to  inquire  what  hap- 
pens in  the  immediate  neighborhood  of  the  opaque  body,  where 
the  laws  are  doubtless  very  complicated  and  where  the  form 
of  the  edge  of  the  screen  must  have  a  perceptible  effect  upon 
the  position  and  the  intensity  of  the  fringes.  We  propose 
rather  to  compute  the  relative  intensities  at  different  points 
of  the  wave-front  only  after  it  has  gone  a  large  number  of  wave- 
lengths beyond  the  screen.  Thus  the  positions  at  which  we 
study  the  waves  are  always  to  be  regarded  as  separated  from 
the  screen  by  a  distance  which  is  very  considerable  compared 
with  the  length  of  a  light-wave. 

34.  We  shall  not  take  up  the  problem  of  vibrations  in  an 
elastic  fluid  from  the  point  of  view  which  the  mathematicians 
have  ordinarily  employed — that  is,  considering  only  a  single 
disturbance.  Single  vibrations  are  never  met  with  in  nature. 
Disturbances  occur  in  groups,  as  is  seen  in  the  pendulum  and 
in  sounding  bodies.  We  shall  assume  that  vibrations  of  lumi- 
nous particles  occur  in  the  same  manner — that  is,  one  after 
another  and  series  after  series.  This  hypothesis  follows  not 
only  from  analogy,  but  as  an  inference  from  the  nature  of  the 
forces  which  hold  the  particles  of  a  body  in  equilibrium.  To 
understand  how  a  single  luminous  particle  may  perform  a  large 
series  of  oscillations  all  of  which  are  nearly  equal,  we  have  only 
to  imagine  that  its  density  is  much  greater  than  that  of  the 
fluid  in  which  it  vibrates — and,  indeed,  this  is  only  what  has 
already  been  inferred  from  the  uniformity  of  the  motions  of 

100 


THE    WAVE-THEORY    OF    LIGHT 

the  planets  through  this  same  fluid  which  fills  planetary  space. 
It  is  not  improbable  also  that  the  optic  nerve  yields  the  sensa- 
tion of  sight  only  after  having  received  a  considerable  number 
of  successive  stimuli. 

However  extended  one  may  consider  systems  of  wave-fronts 
to  be,  it  is  clear  that  they  have  limits,  and  that  in  considering 
interference  we  cannot  predicate  of  their  extreme  portions 
that  which  is  true  for  the  region  in  which  they  are  superposed. 
Thus,  for  instance,  two  systems  of  equal  wave-length  and  of 
equal  intensity,  differing  in  path  by  half  a  wave,  interfere  de- 
structively only  at  those  points  in  the  ether  where  they  meet, 
and  the  two  extreme  half  wave-lengths  escape  interference. 

Nevertheless,  we  shall  assume  that  the  various  systems  of 
waves  undergo  the  same  change  throughout  their  entire  ex- 
tent, the  error  introduced  by  this  assumption  being  inap- 
preciable ;  or,  what  amounts  to  the  same  thing,  we  shall 
assume  in  our  discussion  of  interference  that  these  series  of 
light-waves  represent  general  vibrations  of  the  ether,  and  are 
undefined  as  to  their  limits. 

THE  PROBLEM  OF  INTERFERENCE 

35.  Given  the  intensities  and  relative  positions  of  any  number 
of  trains  of  light-waves  of  the  same  length*  and  travelling  in  the 
same  direction,  to  determine  the  intensity  of  the  vibrations  pro- 
duced by  the  meeting  of  these  different  trains  of  loaves,  that  is, 
the  oscillatory  velocity  of  the  ether  particles.  \ 

*  We  shall  not  here  consider  light-waves  of  different  lengths  which,  in 
general,  come  from  different  sources  and  which  cannot,  therefore,  give 
rise  to  simultaneous  disturbances  and  cannot  by  their  interaction  produce 
any  phenomena  which  are  uniform  ;  and  even  if  they  were  uniform,  the 
rise  and  fall  of  intensity  produced  by  the  interference  of  two  different 
kinds  of  waves,  after  the  manner  of  beats  in  sound,  would  be  far  too 
rapid  to  be  detected,  and  would  produce  only  a  sensation  of  constant  in- 
tensity. ^ 

f  It  was  Mr.  Thomas  Young  who  first  introduced  the  principle  of  inter- 
ference into  optics,  where  he  showed  much  ingenuity  in  applying  it  to 
special  cases;  but  in  the  problems  which  he  has  thus  solved  he  has  con- 
sidered, I  think,  only  the  limiting  cases,  where  the  difference  in  phase  be- 
tween the  two  trains  of  waves  is  either  a  maximum  or  a  minimum,  and  has 
not  computed  the  intensity  of  the  light  for  any  intermediate  cases  or  for 
any  number  whatever  of  trains  of  waves,  as  I  here  propose  to  do. 

101 


MEMOIRS    ON 

Employing  the  general  principle  of  the  superposition  of 
small  motions,  the  total  velocity  impressed  upon  any  particle 
of  a  fluid  is  equal  to  the  sum  of  the  velocities  impressed  by 
each  train  of  waves  acting  by  itself.  When  these  waves  do  not 
coincide,  these  different  velocities  depend  not  only  upon  the 
intensity  of  each  wave,  but  also  upon  its  phase  at  the  instant 
under  consideration.  We  must,  therefore,  know  the  law  ac- 
cording to  which  the  velocity  of  vibration  varies  in  any  one 
wave,  and  for  this  purpose  we  must  trace  the  wave  back  to  the 
origin  whence  it  derives  all  its  characteristics. 

36.  It  is  natural  to  suppose  that  the  particles  whose  vibra- 
tions produce  light  perform  their  oscillations  like  those  of 
sounding  bodies — that  is,  according  to  the  laws  which  hold  for 
the  pendulum ;  or,  what  is  the  same  thing,  to  suppose  that  the 
acceleration  tending  to  make  a  particle  return  to  its  position 
of  equilibrium  is  directly  proportional .  to  the  displacement. 
Let  us  denote  this  displacement  by  x.  A  suitable  function  of 
this  displacement  can  then  be  represented  by  the  expression 
Ax  +  Bx*  +  Cxs-\-etc.,  since  this  will  vanish  when  #=0.  If,  now, 
we  suppose  the  excursion  of  the  particle  to  be  very  small  when 
compared  with  the  radius  of  the  sphere  throughout  which 
the  forces  of  attraction  and  repulsion  act,  we  can  neglect  in 
comparison  with  Ax  all  other  terms  of  the  series  and  con- 
sider the  acceleration  as  practically  proportional  to  the  dis- 
tance x.  This  hypothesis,  to  which  we  are  led  by  analogy, 
and  which  is  the  simplest  that  one  can  make  concerning  the 
vibrations  of  light  particles,  ought  to  lead  to  accurate  results, 
since  the  laws  of  optics  remain  the  same  for  all  intensities  of 
light. 

Let  us  represent  by  v  the  velocity  of  vibration  of  a  light  par- 
ticle at  the  end  of  a  time  t.  We  shall  then  have  dv——Axdt ; 
but  v=dxldt,  or  dt  =  dx/v.  Substituting  in  the  first  equation, 
we  have  vdv——Axdx.  Integrating,  we  have  v*=C—  Ax2;  and 
hence 


A 

Substituting  this  value  of  x  in  the  first  equation,  we  have 

dv 


THE    WAVE-THEORY    OF    LIGHT 

which,  on  integration,  gives 


VA       Vc' 

If  we  measure  time  from  the  instant  at  which  the  velocity 
is  zero,  the  constant  C'  becomes  zero,  and  we  have 
1      .     ,    v 


- or  v= 


If  we  employ  as  unit  of  time  the  interval  occupied  by  the  par- 
ticle in  one  complete  vibration,  we  have  v=  VU  sin  (M). 
Thus,  in  isochronous  vibrations,  the  velocities  for  equal  values 
of  t  are  always  proportional  to  the  constant  V 0,  which,  there- 
fore, measures  the  intensity  of  the  vibration. 

37.  Let  us  now  consider  the  wave  produced  in  the  ether  by 
the  vibrations  of  this  particle.     The  energy  of  motion  in  the 
ether  at  any  point  on  the  wave  depends  upon  the  velocity  of 
the  point-source  at  the  instant  when  it  started  a  disturbance 
which  has  just  reached  this  point.     The  velocity  of  the  ether 
particles  at  any  point  in  space  after  an  interval  of  time  t  is 
proportional  to  that  of  the  point-source  at  the  instant  t—xl\,  x 
being  the  distance  of  this  point  from  the  source  of  motion  and 
\  the  length  of  a  light-wave.     Let  us  denote  by  u  the  velocity 
of  the  ether  particles.     We  then  have 

u—ci  sin 

We  know  that  the  intensity  a  of  vibration*  [oscillatory  ve- 
locity'} in  a  fluid  is  in  inverse  ratio  to  the  distance  of  the  wave 
from  the  centre  of  disturbance;  but,  considering  how  minute 
these  waves  are  when  compared  with  the  distance  which  sep- 
arates them  from  the  luminous  point,  we  may  neglect  the  va- 
riation of  a  and  consider  it  as  constant  throughout  the  extent 
of  one  or  even  of  several  waves. 

38.  By  the  aid  of  this  expression  one  can  compute  the  in- 
tensity of  vibration  produced  by  the  meeting  of  any  number 
of  pencils  of  light  whenever  he   knows  the  intensity  of  the 
different  trains  of  waves  and  their  respective  positions. 

Let  us  first  determine  the  velocity  of  a  luminous  particle  in 
a  vibration  which  results  from  the  interference  of  two  trains 


*  [See  last  sentence  of  section  57  below. ] 
103 


MEMOIRS    ON 

of  waves  displaced,  one  with  respect  to  the  other,  by  a  quar- 
ter of  a  wave-length  [i.e.,  differing  in  phase  ~by  90°],  and  hav- 
ing intensities  which  we  shall  denote  by  a  and  a'.  We  shall 
count  time,  t,  from  the  moment  at  which  the  vibrations  of  the 
first  train  begin.  Let  u  and  u'  be  the  velocities  which  the  first 
and  second  trains  of  waves  would  impress  upon  a  light  particle 
whose  distance  from  the  source  of  motion  is  x.  We  then  have 

u—a  sin    2-*  It—  —  j    and  u'  =  a'  sin  \2-n-  (  4  )  , 

or 


«=-«  -cos 
Hence,  the  resultant  velocity  £7  will  be 

a  sin    ^TT  It  --  I   —  a'  cos    2?r  1  1  --  j. 

Putting  a  =  A  cos  i  and  a'  =  A  sin  i,  this  expression  may  always 
be  placed  in  the  following  form  : 

A  cos  i  sin     2-n-  It  —  —  j  \—A  sin  i  cos    2*  If  —  —  J    , 


or 

A  sin 


in  \2TT  ( ^ )  —  i '  . 


Thus  the  wave  produced  by  the  meeting  of  two  others  will  be 
of  the  same  nature,  but  will  have  a  different  position  [phase] 
and  a  different  intensity.  From  the  equations  A  cos  i=a  and 
A  sin  i=a',  we  have  for  the  value  of  A  (that  is,  for  the  in- 
tensity of  the  resultant  wave)  -v/rt2_j_#'2 ;  but  this  is  exactly  the 
value  of  the  resultant  of  two  mutually  rectangular  forces,  a 
and  a'. 

From  the  same  equations  it  is  easily  seen  also  that  the  new 
wave  exactly  corresponds  in  angular  position  [phase]  to  ,the 
resultant  of  the  two  mutually  rectangular  forces  a  and  a';  for 
the  equation 

U=A  sin 

shows  that  the  linear  displacement  of  this  wave  with  respect 

i\ 
to  the  first  is  — ;  but  i  is  also  the  angle  which  the  force  a 

104 


THE    WAVE-THEORY    OF    LIGHT 

makes  with  the   resultant  A,  because  A  cos  i  —  a.     Thus  we 
have  complete  analogy  between  the  resultant  of  two  mutually 
rectangular  forces  and  the  resultant  of  two  trains  of  waves  dif 
fering  in  phase  by  a  quarter  of  a  wave-length. 

39.  The  solution  of  this  particular  case  for  waves  differing 
by  a  quarter  of  a  wave-length  suffices  to  solve  all  other  cases. 
In  fact,  whatever  be  the  number  of  the  trains  of  waves,  and 
whatever  be  the  intervals  which  separate  them,  we  can  always 
substitute  for  each  of  them  its  components  referred  to  two  ref- 
erence points  which  are  common  to  each  train  of  waves  and  which 
are  distant  from  each  other  by  a  quarter  of  a  wave-length; 
then  adding  or  subtracting,  according  to  sign,  the  intensities  of 
the  components  referred  to  the  same  point,  we  may  reduce  the 
whole  motion  to  that  of  two  trains  of  waves  separated  by  the 
distance  of  a  quarter  of  a  wave-length;  and  the  square  root 
of  the  sum  of  the  squares  of  their  intensities  will  be  the  inten- 
sity of  their  resultant;  but  this  is  exactly  the  method  employ- 
ed in  statics  to  ftrid  the  resultant  of  any  number  of  forces ; 
here  the  wave-length  corresponds  to  one  circumference  in  the 
statical  problem,  and  the    interval   of  a   quarter  of   a  wave- 
length between  the  trains  of  waves  to  an  angular  displacement 
of  90°  between  the  components. 

40.  It  very  often  happens  in  optics  that  the  intensities  of 
light  or  the  particular  tint  which  one  wishes  to  compute  is 
produced  by  the  meeting  of  only  two  trains  of  waves,  as  in  the 
case  of  [Newton's]  colored  rings  and  the  ordinary  phenomena  of 
color  presented  by  crystalline  plates.     It  is,  therefore,  well  to 
know  the  general  expression  for  the  resultant  of  two  trains  of 
waves  differing  in  phase  by  any  amount  whatever.     The  result 
is  easily  predicted   from   the   general  method  which  I  have 
explained,  but  I  think  it  will  be  wise  to  emphasize  somewhat 
the  theory  of  vibrations,  and  to  show  directly  that  the  wave 
resulting  from  two  others,  separated  by  any  interval  whatever, 
corresponds  exactly  in  intensity  and  position  to  the  resultant 
of  two  forces  whose  intensities  are  equal  to  those  of  the  two 
pencils  of  light,  making  an  angle  with  each  other  which  bears 
to  one  complete   circumference  the  same  ratio  that  the  in- 
terval between  the  two  trains   of  waves  bears  to   one  wave- 
length. 

Let  x  be  the  distance  from  the  origin  of  the  first  train  of 
waves  to  the  light  particle  under  consideration,   and   t   the 

105 


MEMOIRS    ON 

instant  for  which  we  wish  to  compute  its  velocity.     The  speed 
impressed  by  the  first  train  of  waves  will  be 


a  sin  [fc  (<-£)], 


where  a  represents  the  intensity  of  this  ray  of  light. 

Let  us  call  a'  the  intensity  of  the  second  pencil,  and  let  us 
denote  by  c  the  distance  between  corresponding  points  on  the 
two  trains  of  waves;  the  [oscillatory]  velocity  due  to  the  second 
train  will  then  be 

a'  sin    %TT  ft—  —  -  j   , 

and  hence  the  total  velocity  impressed  upon  the  particle  will  be 
a  sin    2nl t  —  —  \  \-\-a'  sin   2*1 1 — )   , 


or 


an   expression   to  which   may  always  be  given  the  following 
form : 

A  cos  i  sin    2?r  u  —  -  J    —  A  sin  i  cos    %•*  u  — T  J  \, 
or 

arfn[fc-(<-|)-<], 

where 

#4-^'  cos  (  2?r-  )  =  J  cos  if 

A  V         A/ 

and 

a'  sin  f  2r— |  =  ^4  sin  i. 

Squaring  and  adding,  we  have 

A*=a?-i-a 
Hence, 


A  =  ±\/  ^2  +  ^'3  4-  ###'  cos  ( 2?r  —  J . 

But  this  is  precisely  t*he  value  of  the  resultant  of  two  forces, 
a  and  a',  inclined  to  each  other  at  an  angle  %*—. 

A 

106 


THE    WAVE-THEORY    OF    LIGHT 

41.  From  this  general  expression  it  is  seen  that  the  resultant 
intensity  of  the  light  vibrations  is  equal  to  the  sum  of  intensi- 
ties of  the  two  constituent  pencils  when  they  are  in  perfect 
agreement  and  to  their  difference  when  they  are  in  exactly 
opposite  phases,  and,  lastly,  to  the  square  root  of  the  sum  of 
their  squares  when  their  phase  difference  is  a  quarter  of  a 
wave-length,  as  we'  have  already  shown. 

It  thus  follows  that  the  phase  of  the  wave  corresponds  ex- 
actly to  the  angular  position  of  the  resultant  of  two  forces, 
a  and  a'.  The  distance  from  the  first  wave  to  the  second  is  c, 

to  the  resultant  wave  ^-,  and  from  the  resultant  wave  to  the 

/*7T 

second  is  c—  -—  ;    accordingly,  the  corresponding  angles  are 

(*  C  ' 

2TT.—,  i,  and  2ir.—  —i.     Let  us  multiply  the  equation 

A  A 

«  +  «'  COS  (  2TT  —  \=:A  COS  i 

by  sin  t,  and  the  following  equation 

a'  sin  (  %TT  —  J  =A  sin  i 
by  cos  i.     Subtracting  one  from  the  other,  we  have 

a  sin  i=a'  sin  t  2?r—  —  i  V 
which,  together  with 

a'  sin  (  27T—  j  =  A  sin  i, 

gives  the  following  proportion: 

I  2?r—  —  i\:  sin  i  :  sin  2?r-   :  :  a  :  a'  :  A. 

42.  The  general  expression,  A  sin     gyff—  -Y_t|,  for  the 

velocity  of  the  particles  in  a  wave  produced  by  the  meeting  of 
two  others  shows  that  this  wave  has  the  same  length  as  its 
components  and  that  the  velocities  at  corresponding  points  are 
proportional,  so  that  the  resultant  wave  is  always  of  the  same 
nature  as  its  components  and  differs  only  in  intensity  —  that  is 
to  say,  in  the  constant  by  which  we  must  multiply  the  velocities 
in  either  of  the  components  in  order  to  obtain  the  correspond- 

107 


sn 


MEMOIRS    ON 

ing  velocities  in  the  resultant.  In  combining  this  resultant 
with  still  another  new  wave.,  one  again  arrives  at  an  expression 
of  the  same  form — a  remarkable  property  of  a  function  of  this 
kind.  Thus  in  the  resultant  of  any  number  of  trains  of  waves 
of  the  same  length  the  light  particles  are  always  urged  by  veloc- 
ities proportional  to  those  of  the  components  at  points  located 
at  the  same  distance  from  the  end  of  each  wave.  [This  is  seen 
~by  multiplying  each  of  the  last  three  terms  in  the  preceding  pro- 
portion by  sin  tat.  For  then, 

a  sin  wt :  a1  sin  wi :  A  sin  ut  ::a :  a'  :A' 

: :  constant  ratio.  ] 

APPLICATIONS  OF  HUYGENS'S  PRINCIPLE  TO  THE  PHENOMENA 
OF  DIFFRACTION 

43.  Having  determined  the  resultant  of  any  number  of  trains 
of  light-waves,  I  shall  now  show  how  by  the  aid  of  these  inter- 
ference formulae  and  by  the  principle  of  Huygens  alone  it  is 
possible  to  explain,  and  even  to  compute,  all  the  phenomena 
of  diffraction.  This  principle,  which  I  consider  as  a  rigorous 
deduction  from  the  basal  hypothesis,  maybe  expressed  thus: 
The  vibrations  at  each  point  in  the  wave-front  may  be  considered 
as  the  sum  of  the  elementary  motions  which  at  any  one  instant 
are  sent  to  that  point  from  all  parts  of  this  same  wave  in  any 
one  of  its  previous*  positions,  each  of  these  parts  acting  inde- 
pendently the  one  of  the  other.  It  follows  from  the  principle 
of  the  superposition  of  small  motions  that  the  vibrations  pro- 
duced at  any  point  in  an  elastic  fluid  by  several  disturbances 
are  equal  to  the  resultant  of  all  the  disturbances  reaching  this 
point  at  the  same  instant  from  different  centres  of  vibration, 
whatever  be  their  number,  their  respective  positions,  their 
nature,  or  the  epoch  of  the  different  disturbances.  This  gen- 
eral principle  must  apply  to  all  particular  cases.  I  shall  sup- 
pose that  all  of  these  disturbances,  infinite  in  number,  are  of 
the  same  kind,  that  they  take  place  simultaneously,  that  they 

*I  am  here  discussing  only  an  infinite  train  of  waves,  or  the  most  gen- 
eral vibration  of  a  fluid.  It  is  only  in  this  sense  that  one  can  speak  of  two 
light,  waves  annulling  one  another  when  they  are  half  a  wave-length  apart. 
The  formulae  of  interference  just  given  do  not  apply  to  the  case  of  a  sin- 
gle wave,  not  'o  mention  the  fact  that  such  waves  do  not  occur  in  nature. 

108 


THE    WAVE-THEORY    OF    LIGHT 

are  contiguous  and  occur  in  the  single  plane  or  on  a  single 
spherical  surface.  I  shall  make  still  another  hypothesis  with 
reference  to  the  nature  of  these  disturbances,  viz.,  I  shall  sup- 
pose that  the  velocities  impressed  upon  the  particles  are  all 
directed  in  the  same  sense,  perpendicular  to  the  surface  of  the 
sphere,*  and,  besides,  that  they  are  proportional  to  the  compres- 
sion, and  in  such  a/ way  that  the  particles  have  no  retrograde 
motion.  I  have  thus  reconstructed  a  primary  wave  out  of  par- 
tial [secondary]  disturbances.  We  may,  therefore,  say  that  the 
vibrations  at  each  point  in  the  wave-front  can  be  looked  upon 
as  the  resultant  of  all  the  secondary  displacements  which  reach 
it  at  the  same  instant  from  all  parts  of  this  same  wave  in  some 
previous  position,  each  of  these  parts  acting  independently  one 
of  the  other. 

44.  If  the  intensity  of  the  primary  wave  is  uniform,  it  fol- 
lows from  theoretical  as  well  as  from  all  other  considerations 
that  this  uniformity  will  be  maintained  throughout  its  path, 
provided  only  that  no  part  of  the  wave  is  intercepted  or  re- 
tarded with  respect  to  its  neighboring  parts,  because  the  re- 
sultant of  the  secondary  displacements  mentioned  above  will 
be  the  same  at  every  point.  But  if  a  portion  of  the  wave  be 
stopped  by  the  interposition  of  an  opaque  body,  then  the  in- 
tensity of  each  point  varies  with  its  distance  from  the  edge  of 
the  shadow,  and  these  variations  will  be  especially  marked  near 
the  edge  of  the  geometrical  shadow. 

Let  0  be  the  luminous  point,  AG  the  screen,  AME  a  wave 
which  has  just  reached  A  and  is  partly  intercepted  by  the 
opaque  body.  Imagine  it  to  be  divided  into  an  infinite  num- 
ber of  small  arcs — Am',  m'm,  wM,  Mw,  nn',  n'n",  etc.  In  order 
to  determine  the  intensity  at  any  point  P  in  any  of  the  later  po- 
sitions of  the  wave  BPD,  it  is  necessary  to  find  the  resultant  of 

*It  is  possible  for  light- waves  to  occur  in  which  the  direction  of  the  ab- 
solute velocity  impressed  upon  the  particles  is  not  perpendicular  to  the 
wave  surface.  In  studying  the  laws  of  interference  of  polarized  light,  I 
have  become  convinced  since  the  writing  of  this  memoir  that  light  vibra- 
tions are  at  right  angles  to  the  rays  or  parallel  to  the  wave  surface.  The 
arguments  and  computations  contained  in  this  memoir  harmonize  quite 
as  well  with  this  new  hypothesis  as  with  the  preceding,  because  they  are 
quite  independent  of  the  actual  direction  of  the  vibrations  and  pre-sup- 
pose  only  that  the  direction  of  these  vibrations  is  the  same  for  all  rays 
belonging  to  any  system  of  waves  producing  fringes. 

109 


MEMOIRS    ON 


Fig.  16 


all  the  secondary  waves  which  each  of  these 
portions  of  the  primitive  wave  would  send  to 
the  point  P,  provided  they  were  acting  inde- 
pendently one  of  the  other. 

Since  the  impulse  communicated  to  every' 
part  of  the  primitive  wave  was  directed  along 
the  normal,  the  motion  which  each  [part  of  the 
wave]  tends  to  impress  upon  the  ether  ought 
to  be  more  intense  in  this  direction  than  in 
any  other  ;  and  the  rays  which  would  emanate 
from  it,  if  acting  alone,  would  be  less  and  less 
intense  as  they  deviated  more  and  more  from 
this  direction. 

45.  The  investigation  of  the  law  according  to  which  their 
intensity  varies  about  each  centre  of  disturbance  is  doubtless  a 
very  difficult  matter  ;*  but,  fortunately,  we  have  no  need  of 
knowing  it,  for  it  is  easily  seen  that  the  effects  produced  by 
these  rays  are  mutually  destructive  when  their  directions  are 
sensibly  inclined  towards  the  normal.  Consequently,  the  rays 
which  produce  any  appreciable  Qifect  upon  the  quantity  of 
light  received  at  any  point  P  may  be  regarded  as  of  equal  in- 
tensity, f 

Let  us  now  consider  the  rays  EP,  FP,  and  IP,  which  are  sen- 

*  [This  is  the  problem  solved  by  Stokes;  Math,  and  Phys.  Papers,  vol.  ii.,  p. 
243.] 

f  When  the  centre  of  disturbance  has  been  compressed,  the  force  of  ex- 
pansion tends  to  thrust  the  particles  in  all  directions  ;  and  if  they  have  no 
backward  motion,  the  reason  is  simply  that  their  initial  velocities  forward 
destroy  those  which  expansion  tends  to  impress  upon  them  towards  the 
rear ;  but  it  does  not  follow  that  the  disturbance  can  be  transmitted  only 
along  the  direction  of  the  initial  velocities,  for  the  force  of  expansion  in  a 
perpendicular  direction,  for  instance,  combines  with  a  primitive  impulse 
without  having  its  effect  diminished.  It  is  clear  that  the  intensity  of  the 
wave  thus  produced  must  vary  greatly  at  different  points  of  its  circumfer- 
ence, not  only  on  account  of  the  initial  impulse,  but  also  because  the  com- 
pressions do  not  obey  the  same  law  around  the  centre  of  disturbance  ;  but 
the  variations  of  intensity  in  the  resultant  wave  must  follow  the  law  of 
continuity,  and  may,  therefore,  be  considered  as  vanishing  throughout  a 
small  angle,  especially  along  the  normal  to  the  primitive  wave.  For  the 
initial  velocities  of  the  particles  in  any  direction  whatever  are  proportion- 
al to  the  cosine  of  the  angle  which  this  direction  makes  with  that  of  the 
normal,  so  that  these  components  vary  much  less  rapidly  than  the  angle 
so  long  as  the  angle  is  small. 

110 


THE    WAVE-THEORY    OF    LIGHT 

sibly  inclined  and  which  meet  at  P,  a  point  whose  distance  from 
the  wave  EA  I  shall  suppose  to  include  a  large  number  of  wave- 
lengths. Take  the  two  arcs  EF  and  FI  of  such  a  length  that  the 
differences  EP  — FP  and  FP  — IP  shall  be  equal  to  a  half  wave- 
length. Since  these  rays  are  quite  oblique,  and  since  a  half 
wave-length  is  very  small  compared  with  their  length,  these 
two  arcs  will  be  very  nearly  equal,  and  the  rays  which  they  send 
to  the  point  P  will  be  practically  parallel ;  and  since  corre- 
sponding rays  on  the  two  arcs  differ  by  half  a  wave-length,  the 
two  are  mutually  destructive. 

We  may  then  suppose  that  all  the  rays  which  various  parts 
of  the  primary  wave  AE  send  to  the  point  P  are  of  equal  in- 
tensity, since  the  only  rays  for  which  this  assumption  is  not 
accurate  produce  no  sensible  effect  upon  the  quantity  of  light 
which  it  receives.  In  the  same  manner,  for  the  sake  of  simpli- 
fying the  calculation  of  the  resultant  of  all  the  elementary 
waves,  we  may  consider  their  vibrations  as  taking  place  in  the 
same  direction,  since  the  angles  which  these  rays  make  with 
each  other  are  very  small ;  so  that  the  problem  reduces  itself 
to  the  one  which  we  have  already  solved — namely,  to  find  the 
resultant  of  any  number  of  parallel  trains  of  light-waves  of  the 
same  length,  the  intensities  and  relative  positions  being  given. 
The  intensities  are  here  proportional  to  the  lengths  of  the  il- 
luminating arcs,  and  the  relative  positions  of  the  wave  trains 
are  given  by  the  differences  of  path  traversed. 

46.  Properly  speaking,  we  have  considered  up  to  this  point 
only  the  section  of  the  wave  made  by  a  plane  perpendicular  to 
the  edge  of  the  screen  projected  at  A.  We  shall  now  consider 
it  in  its  entirety,  and  shall  think  of  it  as  divided  by  equidistant 
meridians  perpendicular  to  the  plane  of  the  figure  into  infinitely 
thin  spindles.  We  shall  then  be  able  to  employ  the  same  proc- 
ess of  reasoning  which  we  have  just  used  for  a  section  of  the 
wave,  and  thus  show  that  the  rays  which  are  quite  oblique  are 
mutually  destructive. 

In  the  case  we  are  now  considering  these  spindles  are  indef- 
initely extended  in  a  direction  parallel  to  the  edge  of  the  screen, 
for  the  wave  is  intercepted  only  on  one  side.  Accordingly  the 
intensity  of  the  resultant  of  all  the  vibrations  which  they  send 
to  the  point  P  would  be  the  same  for  each  of  them  ;  for,  owing 
to  the  extremely  small  difference  of  path,  the  rays  which  em- 
anate from  these  spindles  must  be  considered  as  of  equal  in- 
Ill 


MEMOIRS    ON 

tensity,  at  least  throughout  that  region  of  the  primitive  wave 
which  produces  a  sensible  effect  upon  the  light  sent  to  P. 
Further,  it  is  evident  that  each  elementary  resultant  will  differ 
in  phase  by  the  same  quantity  with  respect  to  the  ray  coming 
from  that  point  of  the  spindle  nearest  P,  that  is  to  say,  from 
the  point  at  which  the  spindle  cuts  the  plane  of  the  figure. 
The  intervals  between  these  elementary  resultants  will  then  be 
equal  to  the  difference  of  path  traversed  by  the  rays  AP,  m'P, 
mP,  etc.,  all  lying  in  the  plane  of  the  figure;  and  their  inten- 
sities will  be  proportional  .to  the  arcs  Aw',  m'm,  mM.,  etc.  In 
order  now  to  obtain  the  intensity  of  the  total  resultant,  we 
have  to  -make  the  same  calculation  which  we  have  already 
made,  considering  only  the  section  of  the  wave  by  a  plane  per- 
pendicular to  the  edge  of  the  screen.* 

47.  Before  deriving  the  analytical  expression  for  this  result- 
ant I  propose  to  draw  from  the  principle  of  Huygens  some  of  the 
inferences  which  follow  from  simple  geometrical  considerations. 
Let  AG  represent  an  opaque  body  suffi- 
ciently narrow  for  one  to  distinguish  fringes 
in  its  shadow  at  the  distance  AB.  Let  0  be 
the  luminous  point  and  BD  be  either  the  fo- 
/  i  i  cal  pl^ne  of  the  magnifying-glass  with  which 

one  observes  these  fringes  or  a  white  card 
upon  which  the  fringes  are  projected. 

Let  us  now  imagine  the  original  wave  di- 
vided into  small  arcs — Am,  mm',  m'm",  etc., 
Gn,  nri,  n'n",  ri'ri",  etc. — in  such  a  way 
that  the  rays  drawn  from  the  point  P  in  the 
Fi  shadow  to  two  consecutive  points  of  division 

will  differ  by  half  a  wave-length.  All  of  the 
secondary  waves  sent  to  the  point  P  by  the  elements  of  each 
of  these  arcs  will  completely  interfere  with  those  which  emanate 

*  So  lone:  as  the  edge  of  the  screen  is  rectilinear  we  can  determine  the 
position  of  the  dark  and  bright  bands  and  their  relative  intensities  by  con- 
sidering only  the  section  of  the  wave  made  by  a  plane  which  is  perpendic- 
ular to  the  edge  of  the  screen.  But  when  the  edge  of  the  screen  is  curved 
or  composed  of  straight  edges  inclined  at  an  angle  it  is  then  necessary  to 
integrate  along  two  directions  at  right  ansrles  to  each  other,  or  to  integrate 
around  the  point  under  consideration.  In  some  particular  cases  this  latter 
method  is  simpler,  as,  for  instance,  when  we  have  to  calculate  the  intensity 
of  the  light  in  the  centre  of  the  shadow  produced  by  a  screen  or  in  the 
projection  of  a  circular  aperture. 

112 


THE    WAVE-THEORY    OF    LIGHT 

from  the  corresponding  parts  of  the  two  arcs  immediately  ad- 
joining it  "^  so  that,  if  all  these  arcs  were  equal,  the  rays  which 
they  would  send  to  the  point  P  would  be  mutually  destructive, 
with  the  exception  of  the  extreme  arc  mA.  Half  of  the  in- 
tensity of  this  arc  would  be  left,  for  half  the  light  sent  by 
the  arc  mm'  (with  which  mA  is  in  complete  discordance) 
would  be  destroyed,  by  half  of  the  preceding  arc  m"m'.  As 
soon  as  the  rays  meeting  at  P  are  considerably  inclined  with 
respect  to  the  normal,  these  arcs  are  practically  equal.  The 
resultant  wave,  therefore,  corresponds  in  phase  almost  ex- 
actly to  the  middle  of  mA,  the  only  arc  which  produces 
any  sensible  effect.  It  is  thus  seen  that  it  differs  in  phase 
by  one-quarter  of  a  wave-length  from  the  element  at  the  edge 
A  of  the  opaque  screen.  Since  the  same  thing  takes  place  in 
the  other  part  of  the  incident  wave  Qn,  the  interference  be- 
tween these  two  vibrations  occurring  at  the  point  P  is  deter- 
mined by  the  difference  of  length  between  the  two  rays  sP  and 
tP,  which  take  their  rise  at  the  middle  of  the  arcs  Am  and  Gn, 
or,  what  amounts  to  the  same  thing,  by  the  difference  between 
the  two  rays  AP  and  GP  coming  from  the  very  edge  of  the 
opaque  body.  It  thus  happens  that  when  the  interior  fringes 
under  consideration  are  rather  distant  from  the  edges  of  the 
geometrical  shadow,  we  are  able  to  apply  practically  without 
error  the  formula  based  upon  the  hypothesis  that  the  inflected 
waves  have  their  origin  at  the  very  edges  of  the  opaque  body; 
but  in  proportion  as  the  point  P  approaches  B  the  arc  Am  be- 
comes greater  in  comparison  with  the  arc  mm',  the  arc  mm' 
with  respect  to  the  arc  m'm",  etc.  ;  and  likewise  in  the  arc  mA 
the  elements  in  the  immediate  vicinity  of  the  point  A  become 
sensibly  greater  than  the  elements  which  are  situated  near  the 
point  m,  and  which  correspond  to  equal  differences  of  path.  It 
happens,  therefore,  that  the  effective*  ray,  sP,  will  not  be  the 
mean  between  the  outside  rays,  mP  and  AP,  but  will  more 
nearly  approach  the  length  of  the  latter.  On  the  other  side 
of  the  opaque  body  we  have  slightly  different  circumstances. 
The  difference  between  the  ray  GP  and  the  effective  ray  tP  ap- 
proximates more  and  more  nearly  a  quarter  of  a  wave-length 

*  I  have  given  this  name  to  the  distance  of  the  resultant  wave  from  the 
original  wave  because  the  positions  of  the  dark  and  bright  bands  are  the 
same  as  they  would  be  if  these  effective  rays  alone  produced  them. 
H  113 


MEMOIRS    ON 


as  the  point  P  moves  farther  and  farther  away  from  D,  so  that 
the  difference  of  path  traversed  varies  more  rapidly  between 
the  effective  rays  sP  and  tP  than  between  the  rays  AP  and  GP  ; 
consequently,  the  fringes  in  the  neighborhood  of  the  point  B 
ought  to  be  a  little  farther  from  the  centre  of  the  shadow  than 
would  be  indicated  by  the  formula  based  upon  the  first  hy- 
pothesis. 

48.  Having  considered  the  case  of  fringes  produced  by  a 
narrow  body,  I  pass  to  the  consideration  of  those  which  are 
caused  by  a  small  aperture. 

Let  AG  be  the  aperture  through  which 
the  light  passes.  I  shall  at  first  suppose 
that  it  is  sufficiently  narrow  for  the  dark 
bands  of  the  first  order  to  fall  inside  the 
geometrical  shadow  of  the  screen,  and 
at  the  same  time  to  be  fairly  distant 
from  the  edges  B  and  D.  Let  P  be  the 
darkest  point  in  one  of  these  two  bands; 
it  is  then  easily  seen  that  this  must  cor- 
respond to  a  difference  of  one  whole  wave- 
length between  the  two  extreme  rays  AP 
and  GP.  Let  us  now  imagine  another 
ray,  PI,  drawn  in  such  a  way  that  its 
length  shall  be  a  mean  between  the  other 
two.  Then,  on  account  of  its  marked  in- 
clination to  the  arc  AIG,  the  point  I  will 
fall  almost  exactly  in  the  middle.  We  now  have  the  arc  di- 
vided into  two  parts,  whose  corresponding  elements  are  almost 
exactly  equal,  and  send  to  the  point  P  vibrations  in  exactly 
opposite  phases,  so  that  these  must  annul  each  other. 

By  the  same  reasoning  it  is  easily  seen  that  the  darkest 
points  in  the  other  dark  bands  also  correspond  to  differences 
of  an  even  number  of  half  wave-lengths  between  ihe  two  rays 
which  come  from  the  edges  of  the  aperture ;  and,  in  like  man- 
ner, the  brightest  points  of  the  bright  bands  correspond  to 
differences  of  an  uneven  number  of  half  wave-lengths — that  is 
to  say,  their  positions  are  exactly  reversed  as  compared  with 
those  which  are  deduced  from  the  interference  of  the  limiting 
rays  on  the  hypothesis  that  these  alone  are  concerned  in  the 
production  of  fringes.  This  is  true  with  the  exception  of  the 
point  at  the  middle,  which,  on  either  hypothesis,  must  be 

114 


THE    WAVE-THEORY    OF 

bright.  The  inferences  deduced  from  the  theory 
fringes  result  from  the  superposition  of  all  of  the  disturbances 
from  all  parts  of  the  arc  AGr  are  verified  by  experiments, 
which  at  the  same  time  disprove  the  theory  which  looks  upon 
these  bands  as  produced  only  by  rays  inflected  and  reflected  at 
the  edges  of  the  diaphragm.  These  are  precisely  the  phenom- 
ena which  first  led  me  to  recognize  the  insufficiency  of  this 
hypothesis,  and  suggested  the  fundamental  principle  of  the 
theory  which  I  have  just  explained — namely,  the  principle  of 
Huygens  combined  with  the  principle  of  interference. 

49.  In  the  case  which  we  have  just  considered,  where,  by 
virtue  of  a  very  small  aperture,  the  dark  bands  of  the  first  or- 
der fall  at  some  distance  from  the  edges  of  the  geometrical 
shadow,  it  follows  from  theory,  as  well  as  from  experiment, 
that  the  distance  comprised  between  the  darkest  points  is  al- 
most exactly  double  that  of  the  other  intervals  between  the 
middle  points  of  two  consecutive  dark  bands,  and  this  is  all 
the  more  nearly  true  in  proportion  as  the  aperture  becomes 
smaller  or  more  distant  from  the  luminous  point  and  from  the 
focus  of  the  magnifying-glass  with   which   one   observes   the 
fringes ;  for,  by  sufficiently  increasing  these  distances  one  may 
produce  the   same  effects  with  an  aperture  of  any  size  what- 
ever. 

But  when  these  distances  are  not  very  great,  and  when  their 
aperture  is  too  large  for  the  rays  producing  the  fringes  to  be 
very  much  inclined  to  the  wave-front,  AG-,  it  follows  that 
corresponding  elements  of  the  arcs  into  which  we  have  sup- 
posed a  wave  to  be  divided  can  no  longer  be  considered  as  each 
equal  to  the  other,  for  they  are  sensibly  larger  on  the  side  next 
the  band  un'der  consideration.  Under  these  conditions  we 
can  rigorously  deduce  the  positions  of  maximum  and  mini- 
mum intensity  only  by  computing  the  resultant  of  all  the 
small  secondary  waves  which  are  sent  out  by  the  incident 
wave. 

50.  But  there  is  one  very  remarkable  case  where  a  knowl- 
edge  of  this  integral  is  not  needed   for  the  determination  of 
the  law  of  the  fringes  by  an  aperture  of  very   considerable 
size.     This  is  the  case  where  a  lens  is  placed  in  front  of  the 
diaphragm,  and   brings  the  refracted  rays  to  focus   upon  the 
plane  in  which  the  fringes  are  observed.     The  problem  is  now 
greatly  simplified  by  the  fact  that  the  centre  of  curvature  of  the 

115 


MEMOIRS    ON 


emergent  wave  now  lies  in  this  plane  instead  of  at  the  lumi- 
nous point. 

Let  0  be  the  projection  of  the  middle 
point  of  the  aperture  upon  this  plane. 
From  the  point  0  as  centre,  and  with  a 
radius  equal  to  AO,  let  us  now  describe 
the  arc  AI'Gr,  which  will  now  represent 
the  incident  wave  as  modified  by  the  inter- 
position of  the  lens.  If,  now,  from  the 
point  P  as  centre,  and  with  a  radius  AP, 
we  describe  the  arc  AEF,  those  portions 
of  the  luminous  rays  meeting  at  the  point 
P  which  are  comprised  between  the  arc 
AI'G-  and  the  arc  AEF  will  be  the  differ- 
ences of  path  traversed  by  the  secondary 
waves ;  and,  since  these  two  arcs  have 
equal  curvatures  and  are  convex  towards 
the  same  side,  it  follows  that  equal  differ- 
ences of  path  will  correspond  to  equal  intervals  upon  the  wave- 
front  AI'G.  Let  us  suppose  this  wave  divided  in  such  a  manner 
that  any  two  consecutive  rays  drawn  through  the  points  of  di- 
vision shall  differ  by  one-half  a  wave-length.  If,  then,  the  point 
P  be  located  in  such  a  way  that  the  total  number  of  these  arcs  is 
even,  it  will  no  longer  receive  any  light.  For  these  arcs,  taken 
two  and  two,  are  mutually  destructive,  since  the  vibrations  due 
to  corresponding  elements  are  at  the  same  time  of  equal  in- 
tensity and  opposite  phase.  The  light  reaching  any  point 
P  will  be  a  maximum  when  the  total  number  of  arcs  is  un- 
even. The  brightest  points  of  the  bright  bands,  therefore,  cor- 
respond to  a  difference  of  an  uneven  number  of  half  wave- 
lengths between  the  two  rays  coming  from  the  edges  of  the 
diaphragm,  and  the  darkest  points  on  the  dark  bands  to  a  dif- 
ference of  an  even  number  of  half  wave-lengths.  Consequently, 
all  the  dark  bands  will  be  equally  spaced  among  themselves, 
with  the  exception  of  the  first  two,  where  the  interval  is  ex- 
actly double  that  which  separates  the  others.  This  result, 
which  had  already  been  suggested  by  theory,  I  found  to  be 
thoroughly  confirmed  by  experiment.  I  shall  cite  only  one 
experiment  of  this  kind  made  in  homogeneous  red  light.  In 
order  to  bring  the  centre  of  the  incident  wave  to  the  plane  of 
the  micrometer  wire,  I  used,  instead  of  an  ordinary  lens,  a 

116 


THE    WAVE-THEORY    OF    LIGHT 

glass  cylinder,  which,  in  order  to  get  the  full  length  of  the 
fringes,  I  placed  with  its  generating  line  parallel  to  the  edges 
of  the  aperture  in  the  diaphragm. 

mm. 

Size  of  the  aperture 2.00 

m. 

Distance  from  the  luminous  point  to  the  diaphragm,  or  a 2  507 

Distance  from  the  diaphragm  to  the  micrometer,  or  b 1.140 

mm. 
Interval  between  the  middle  points  of  the  two  dark  bands  of  the  first 

order 0.72 

Interval  between  the  band  of  the  first  order  and  the  third 0.73 

Interval  between  the  band  of  the  third  order  and  the  fifth 0.72 

It  will  be  observed  that  the  first  interval  is  double  that  of 
the  others. 

I  have  observed  that  the  same  law  holds,  even  at  distances 
which  are  not  very  great,  for  apertures  which  are  much  wider, 
a  centimeter  or  even  a  centimeter  and  a  half  ;  but  if  we  further 
increase  the  aperture  of  the  diaphragm,  the  fringes  become 
confused,  however  much  care  be  taken  to  place  the  microme- 
ter in  the  focus  of  the  cylindrical  lens;  which  goes  to  show 
that  the  rays  refracted  by  this  glass  vibrate  in  unison  [in  the 
same  phase}  only  within  rather  narrow  limits,  just  as  happens 
with  ordinary  lenses. 

51.  When  the  aperture  of  the  diaphragm  thus  backed  with  a 
cylindrical  lens  is  not  too  great,  the  dark  and  bright  bands 
produced  are  as  sharp  as  the  fringes  which  result  from  the 
union  of  rays  reflected  from  two  mirrors.  But,  in  the  latter 
case,  the  intensity  of  the  light  is  the  same  for  all  fringes,  or,  at 
least,  whatever  differences  there  are  appear  to  arise  merely  from 
the  fact  that  the  light  employed  is  not  perfectly  homogeneous ; 
and  if  it  happen  that  the  bright  bands  diminish  in  brilliancy, 
the  dark  bands  become  less  dark,  so  that  the  sum  of  the  light 
in  one  entire  fringe  remains  practically  constant.  But  in  the 
other  phenomenon,  as  one  recedes  from  the  centre  he  observes 
a  rapid  diminution  of  the  light,  which  is  easily  accounted  for 
by  the  theory  we  have  just  explained.  For,  indeed,  all  the 
rays  which  leave  the  wave-front  AI'Gr  and  meet  at  the  centre 
of  the  bright  band  of  the  first  order  have  traversed  equal 
paths  ;  so  that  all  the  small  secondary  waves  which  they  bring 
to  this  point  coincide  \in  phase]  and  strengthen  each  other. 

117 


MEMOIRS    ON 

But  this  is  not  the  case  with  the  other  bright  bands.  The 
brightest  band  of  the  second  order,  for  instance,  corresponds 
to  a  division  of  the  wave  AI'G-  into  three  arcs,  the  extreme 
rays  of  which  differ  by  one-half  a  wave-length  ;  the  effects 
produced  by  two  of  these  arcs  annul  each  other.  Consequent- 
ly, this  band  receives  light  from  only  one-third  of  the  incident 
wave-front,  while  even  the  effect  produced  by  this  third  is 
somewhat  diminished  by  the  fact  that  there  is  a  difference  of 
one-half  a  wave-length  between  the  rays  from  its  edges.  A 
similar  process  of  reasoning  shows  that  the  middle  of  the 
bright  band  of  the  third  order  is  illuminated  by  only  one-fifth 
of  the  wave-front  AI'G,  the  light  of  this  one-fifth  being  still 
further  diminished  by  opposition  of  phase  in  its  extreme 
rays. 

[Here  are  omitted  six  pages,  including  a  geometrical  discussion 
of  the  general  relations  between  size  of  aperture  (or  obstacle),  dis- 
tance of  screen,  distance  of  luminous  point,  etc.] 

56.  I  have  just  explained  the  general  relations  between  the 
size  of  any  particular  fringe  and  the  respective  distances  of  the 
obstacle  from  the  luminous  point  and  from  the  micrometer. 
As  we  have  seen,  these  laws  may  be  derived  from  theory  quite 
independently  of  any  knowledge  of  the  integral  which  at  each 
point  represents  the  resultant  of  all  the  secondary  waves  ;  but 
in  order  to  find  the  absolute  size  of  these  fringes,  it  is  essential 
that  we  compute  this  resultant,  for  the  positions  of  maxima  and 
minima  of  intensity  can  be  determined  only  by  a  comparison 
of  the  different  values  of  this  resultant,  or  at  least  by  knowing 
the  function  which  represents  it. 

In  order  to  do  this,  we  propose  to  apply  to  the  principle  of 
Huygens  the  method  which  we  have  already  explained  for  com- 
puting the  resultant  of  any  number  of  trains  of  waves  when 
their  intensities  and  relative  positions  [phases']  are  given. 

APPLICATION   OF  THEORY  OF  INTERFERENCE  TO   HUYGENS'S 

PRINCIPLE 

57.  Let  the  waves  from  any  luminous  point  0  be  partly  inter- 
cepted by  an  opaque  body  AG.     To  begin  with,  we  shall  sup- 
pose that  this  screen  is  so  large  that  no  light  comes  around  the 
edge  G,  so  that  we  need  consider  only  that  part  of  the  wave 
which  lies  to  the  left  of  the  point  A.     Let  DB  represent  the 

118 


THE    WAVE-THEORY    OF    LIGHT 

plane  upon  which  are  received  the  shadow  and  its  fringes.  The 
problem  then  is  to  find  the  intensity  .of  the  light  at  any  point 
P  in  this  plane. 

If  from  C  as  centre  and  with  a  radius  CA  we  describe  the 
circle  AMI,  it  will  represent  the  light- wave  at  the  instant  it 
is  partly  intercepted  by  the  opaque  body. 
It  is  from  this  position  of  the  wave  that  I 
have  computed  the  resultant  of  the  sec- 
ondary waves  sent  to  the  point  P..  If  we 
consider  the  wave  in  an  earlier  position, 
say  A'MT,  i.t  then  becomes  necessary  to 
calculate  the  effect  of  the  obstacle  on  each 
of  the  secondary  waves  arising  from  the 
arc  A'MT  ;  and  if  we  consider  the  wave 
in  a  later  position,  say  A"M"I",  it  becomes 
necessary  to  first  determine  the  intensities 
of  its  various  points,  for  they  are  no  longer 
equal,  having  been  changed  by  the  inter- 
position of  the  screen.  In  this  case  the 
computation  is  vastly  more  complicated, 
possibly  quite  impracticable.  If,  however, 
we  consider  the  wave  at  the  instant  it 


Fig.  19 


reaches  A,  the  process  is  simple  ;  for  then  all  parts  of  the  wave 
have  the  same  intensity.  Not  only  so,  but  none  of  the  second- 
ary waves  are  now  affected  by  the  opaque  screen.  However 
numerous  the  subdivisions  into  which  we  may  consider  these 
elementary  waves  divided,  it  is  evident  that  the  number  will 
be  the  same  for  each,  since  they  are  transmitted  freely  in  all 
directions.  And,  therefore,  we  need  only  consider  the  axes  of 
these  pencils  of  split  rays—?,  e.,  the  straight  lines  drawn  from 
the  various  points  on  the  wave  AMI  to  the  point  P.  The  dif- 
ferences of  length  in  these  direct  rays  are  the  differences  of 
path  traversed  by  the  elementary  or  partial  resultants  meeting 
at  P.* 

In  order  to  compute  the  total  effect,  I  refer  these  partial  re- 
sultants to  the  wave  emitted  by  the  point  M  on  the  straight  line 
CP,  and  to  another  wave  displaced  a  quarter  of  a  wave-length 
with  reference  to  the  preceding.  This  is  the  process  already 
employed  (p.  101)  in  the  general  solution  of  the  interference 


*  [Afoot-note  is  here  omitted.'] 
119 


MEMOIRS    ON 

problem.  We  shall  consider  only  a  section  of  the  wave  made 
by  a  plane  perpendicular  to  the  edge  of  the  screen,  and  shall 
indicate  by  dz  an  element,  nn',  of  the  primary  wave,  and  by  z  its 
distance  from  the  point  M.  These,  as  I  have  shown,  suffice  to 
determine  the  position  and  the  relative  intensities  of  the  bright 
and  dark  bands.  The  distance  nS  included  between  the  wave 
AMI  and  the  tangential  arc,  EMF,  described  about  the  point  P 

as  centre  is  |  — — =— ^  where. a  and  b  are,  as  before,  the  distances 
ab 

CA  and  AB.  If  we  denote  the  wave-length  by  X,  we  have  for 
the  component  in  question,  referred  to  the  wave  leaving  the 
point  M,  the  following  expression 

dz  cos  (  TT          — - )  ; 
\        abX    J  ' 

while  for  the  other  component,*  referred  to  a  wave  displaced  a 
quarter  of  a  wave-length  from  the  first,  we  have 


If,  now,  we  take  the  sum  of  all  similar  components  of  all  the 
other  elements,  we  shall  have 


rdz  cos  ,    »+    and  rdz  siri 

J  ab\     I          J 


ab\ 


Hence  the  intensity  of  the  vibration  at  P  resulting  from  all 
these  small  disturbances  is 


The   intensity  of  the  sensation,  being  proportional  to  the 
square  of  the  speeds  of  the  particles,  is 


This  is  what  I  have  called  the  intensity  of  the  light  in  order  to 
conform  to  ordinary  usage,  while  reserving  the  expression  in- 
tensity of  vibration  to  designate  the  speed  of  an  ether  particle 
during  its  oscillation. 

*  [  TJiese  expressions  for  amplitude  follow  directly  from  sec.  40,  when  in  the 

ke  a. 
120 


general  expression  for  velocity  we  make  a—o,  a'=dz,  and  c=^-— — .] 


THE    WAVE-THEORY    OF    LIGHT 

58.  In  the  case  we  are  now  considering,  where  the  body,  AG, 
is  so  large  that  we  can  neglect  any  light  coming  around  the 
edge  G,  the  integration  extends  from  A  to  infinity  on  the  side 
towards  I.  This  integral  naturally  divides  into  two  parts,  one 
extending  from  A  to  M,  the  other  from  M  to  infinity.  This 
latter  integral  remains  constant,  while  the  former  varies  with 
the  position  of  the  point  P.  This  variation,  indeed,  is  the  de- 
termining factor  in  the  size  and  relative  intensity  of  the  bright 
and  dark  bands. 

The  integrals 

r,          /    z*(a+b)\       -,/»,. 
Jdz  cos  (T-^)  and  /  dz  sin 

may  be  evaluated  in  finite  terms  when  the  limits  of  z  are  taken 
at  zero  and  infinity ;  but  between  any  other  limits  their  values 
can  be  expressed  only  in  terms  of  a  series  or  by  means  of  par- 
tial integration. 

The  latter  method  seems  to  me  more  convenient,  and  I  have, 
therefore,  employed  it  in  the  computation  of  the  following 
table,  where  the  limits  of  integration  are  taken  so  close  together 
that  we  can  neglect  the  square  of  half  the  arc  included  between 
them.* 


*  Let  *  and  i+t  be  the  narrow  limits  between  which  it  is  proposed  to  in- 

tegrate dv  cos  q^  and  dv  sin  qv'2.     Neglecting  the   square  of  -,  we  then 

& 
timl  the  following  approximate  values  for  these  integrals: 


These  are  the  formulae  which  I  have  used  in  the  computation  of  the  table. 
When  the  limits  are  sufficiently  narrow  for  us  to  neglect  <2  instead  of 

5  j  j  the  following  still  simpler  formnlae  may  be  employed  : 
/  dv  cos  9^=2^  sin  qi  (i+2t)-sin  qi* 


i+t 

/**  i  r 

I  dv  sin  qv*=p-T-\  —cos  otf  (i 
/  2iq\ 

•/t+t  L 

121 


MEMOIRS    ON 

This  arc  here  amounts  to  ^  of  a  quadrant,  since  this  fur- 
nishes results  of  an  accuracy  greater  than  is  attainable  in  the 
observations.  In  place  of  the  integrals  mentioned  above,  I  have 
substituted  fdv  sin  qv"1  and  fdv  cos  qv*,  where  q  stands  for 

quadrant  or  -. 

/v 

To  pass  from  one  of  these  forms  to  the  other  is  a  simple 
matter. 


[Following  is  a  derivation  of  these  formulae,  which  Verdet  found  in  one  of 
Fresnel's  journals. 

Let  the  limits  of  integration  he  denoted  by  a  and  a  -f  2p. 

Put  v=a+p-\-u.  Then  du=dv  ;  and  when  v=a,  u=—p;  but  when 
v=a+2p,  u=+p. 

Substituting  for  v, 

I  dv  cos  qv*=  I  du  cos  q\  u'1  +  2(a  +p)u  +  (a  +_^»)2  1. 

If  the  limiting  values  of  u  are  +p  and  —p,  we  may  take  p  so  smalL  say 
Y1^,  that  we  may  neglect  its  square,  u1.  We  then  have 

Ca  i*+p        \  1 

I  dv  cos  qv*=  I  du  cos  q\  Zu(a+p)+(a+p)* 

Ja+*P  J  -p 


2q(a+p) 

122 


THE    WAVE-THEORY    OF    LIGHT 


TABLE    OF   THE   NUMERICAL   VALUES    OF   THE    INTEGRALS 

fdv  cos  qv*  and  J  dv  sin  qv*.* 


Limits 
of 
Integrals 

J'dv  cos  qv* 

J'dv  sin  gt>3 

Limits 
of 
Integrals 

fdv  cos  5«a 

J'dv  sin  qv* 

From  v=W 

From  v=W 

to        fl=(K10 

0.0999 

0.0006 

to        0=2*.  90 

0.5627 

0.4098 

to        «=0.20 

0.1999 

0.0042 

to             3.00 

0.6061 

0.4959 

0.30 

0.2993 

0.0140 

3.10 

0.5621 

0.5815 

0.40 

0.3974 

0.0332 

3.20 

0.4668 

0.5931 

0.50 

0.4923 

0.0644 

3.30 

0.4061 

0.5191 

0.60 

0.5811 

0.1101 

3.40 

0.4388 

0.4294 

0.70 

0.6597 

0.1716 

3.50 

0.5328 

0.4149 

0.80 

0.7230 

0.2487 

3.60 

0.5883 

0.4919 

0.90 

0.7651 

0.3391 

3.70 

0.5424 

0.5746 

1.00 

0.7803 

0.4376 

3.80 

0.4485 

0.5654 

1.10 

0.7643 

0.5359 

3.90 

0.4226 

0.4750 

1.20 

0.7161 

0.6229 

4.00 

0.4986 

0.4202 

1.30 

0.6393 

0.6859 

4.10 

0:5739 

0.4754 

1.40 

0.5439 

0.7132 

4.20 

0.5420 

0.5628 

1.50 

0.4461 

0.6973 

4.30 

0.4497 

0.5537 

1.60 

0.3662 

0.6388 

4.40 

0.4385 

0.4620 

1.70 

0.3245 

0.5492 

4.50 

0.5261 

0.4339 

1.80 

0.3342 

0.4509 

4.60 

0.5674 

0.5158 

1.90 

0.3949 

0.3732 

4.70 

0.4917 

0.5668 

2.00 

0.4886 

0.3432 

4.80 

0.4340 

0.4965 

2.10 

0.5819 

0.3739 

.4.90 

0.5003 

0.4347 

2.20 

0.6367 

0.4553 

5.00 

0.5638 

0.4987 

2.30 

0.6271 

0.5528 

5.10 

0.5000 

0.5620 

2.40 

0.5556 

0.6194 

5.20 

0.4390 

0.4966 

2.50 

0.4581 

0.6190 

5.30 

0.5078 

0.4401 

2.60 

0.3895 

0.5499 

5.40 

0.5573 

0.5136 

2.70 

0.3929 

0.4528 

5.50 

0.4785 

0.5533 

2.80 

0.4678 

0.3913 

*  From  the  text,  and  also  from  the  first  column  of  this  table,  one  would 
be  led  to  think  that  the  second  and  third  columns  in  the  table  give  the 


values  of  the 
values  of  v : 


integrals  /  a 

v  o 


dv  cos  —  V2  and 


/• 
dv  sin  - 


for  the  following 


etc. 

That  this  is  not  the  case,  however,  may  be  shown  by  using  the  approxima- 
tion formulae  of  Fresnel  to  compute  any  pair  of  consecutive  values  of 

123 


MEMOIRS    ON 

Either  of  the  integrals   i  dv  cos  qv*  and  jdv  sin  qv*  taken 

from  zero  to  infinity  have  the  value  -3-.  We  may  thus  by  the 
aid  of  the  above  table  find  the  intensity  of  light  corresponding 
to  any  given  position  of  the  point  P,  or,  what  is  the  same 
thing,  corresponding  to  any  definite  value  of  v,  where  v  is  one 
limit  of  integration  and  infinity  the  other.  We  have  only  to 

take  from  the  table  the  values  of  J  civ  cos  qv*  and  /  dv  sin  qv*, 

using  the  value  of  v  as  an  argument,  then  add  to  each  £,  and 
finally  take  the  sum  of  their  squares. 

59.  Simple  inspection  of  this  table  shows  a  periodic  change 
in  the  intensity  of  light  as  one  leaves  the  geometrical  shadow. 
To  obtain  the  values  of  v  corresponding  to  maxima  and  minima, 
i.  e.,  the  brightest  and  darkest  points  in  the  respective  bright 
and  dark  bands,  I  take  from  the  table  the  numbers  which  most 
nearly  correspond  to  them  and  then  compute  the  correspond- 
ing intensities.  Finally,  by  means  of  these  data  and  a  simple 
formula  of  approximation,  I  determine  with  sufficient  accu- 
racy the  values  of  v.  which  give  maxima  and  minima. 

Let  us  represent  by  *  the  approximate  value  of  v  taken  di- 
rectly from  the  table,  by  /  and  Y  the  corresponding  values  of 

£_[_  jdv  cos  qv*  and  i+jdv  sin  qv*,  and  by  t  the  small  arc  by 

which  v  must  be  increased  in  order  to  give  the  maximum  or 
minimum  of  light.  Neglecting  the  square  of  t,  we  find  that 
the  following  formula  gives  the  value  of  t  which  yields  a 
maximum  or  a  minimum. 

f   /-a,  o'Al-  %qil—  sin  qi*  _ 

n    £    i       A.       - 


[A  foot-note  containing  the  derivation  of  this  expression  is 
here  omitted.} 

If  in  this  formula  we  substitute  the  numbers  taken  from  the 
table,  we  obtain  the  following  results  : 

either  integral.     The  successive  values  of  v  employed  in  the  first  column 
are 

6=0.1, 

0=0.2, 
v=0.3, 

etc. 
The  same  remark  applies  to  the  following  tables.  [E.  Verdet.] 

124 


THE    WAVE-THEOKY    OF    LIGHT 

TABLE     OF     MAXIMA    AND     MINIMA     FOR     EXTERIOR     FRINGES 
AND    OF   THE    CORRESPONDING    INTENSITIES 


Values  of  V 

Intensities 
of  Light 

Maximum  of  1st  order  

1  2172 

2  7413 

Minimum  of  1st  order 

1  8726 

1  5570 

Maximum  of  2d  order.    .  .    . 

2  3449 

2  3990 

Minimum  of  3d  order  

2  7392 

1  6867 

Maximum  of  3d  order 

3  0820 

2  3022 

Minimum  of  3d  order  

3  3913 

1  7440 

Maximum  of  4th  order 

3  6742 

2  2523 

Minimum  of  4th  order  

3  9372 

1  7783 

Maximum  of  5th  order  

4  1832 

2  2206 

Minimum  of  5th  order 

44160 

1  8014 

Maximum  of  6th  order 

4  6369 

2  1985 

Minimum  of  6th  order  

4  8479 

1  8185 

Maximum  of  7th  order 

5  0500 

2  1818 

Minimum  of  7th  order        .  .   . 

5  2442 

1  8317 

It  is  to  be  observed  that  here  none  of  the  minima  become 
zero,  as  in  the  case  of  Newton's  rings,  or  in  fringes  produced 
by  the  meeting  of  two  beams  of  light  of  equal  intensities  ;  here 
the  difference  between  maxima  and  minima  diminishes  as  one 
goes  farther  away  from  the  edge  of  the  opaque  screen. 

This  explains  why  the  fringes  which  border  shadows  are  not 
so  bright  or  so  numerous  as  the  colored  rings,  or  as  the  bands 
produced  by  the  reflection  of  a  luminous  point  in  two  slightly 
inclined  mirrors. 

60.  To  employ  the  above  table  in  computing  the  size  of  the 
exterior  fringes,  we  must  first  recall  the  substitution  of  the  in- 
tegrals /  dv  cos  qv*  and  /  dv  sin  qv*  for  the  integrals  *  in 
question, 


cos    - 


_     . 

(fesin 


whence 


and 


[In  what  follows  Fresnel  replaces  =  by  q.  the  initial  letter  of  "quadrant."] 

125 


MEMOIRS    ON 
Therefore 

fdz  cos  fa  ^+*>)  .__  /~^2  fdv  cos 
J  \          ab\     I     V  2(^fS)J 

and 


Also 


c" 


Now  the  factor  ^  —  —rr-  is  constant  ;   whence  we  infer  that 
4{a+o) 

these  two  quantities, 


and 

/  r  ,  A2     /  r  , 

sin 


I  I  dv  cos  gt^J  4-  (  ^ 


will  each  reach  their  maximum  or  minimum  values  at  the  same 
time.  Let  us  now  denote  by  n  the  value  of  v  which  yields  a 
maximum  or  minimum  value  for  these  integrals  ;  the  corre- 
sponding value  of  z  will  then  be 


„     „.   ,      ##X 

Z^r 


The  size  of  the  fringe,  x,  [that  is,  its  distance  from  the  edge  of 
the  opaque  screen]  then  follows  from  the  proportion 


a  :  z  :: 
whence 


or,  substituting  for  z, 


This  radical,  it  may  be  remarked,  is  exactly  the  distance 
between  the  edge  of  the  geometrical  shadow  and  that  point 

126 


THE    WAVE-THEORY    OF    LIGHT 

which  corresponds  to  a  difference*  of  a  quarter  of  a  wave-length 
between  the  direct  ray  and  the  ray  coming  via  the  edge  of  the 
opaque  screen.  But  this  is  precisely  what  might  have  been 
predicted,  inasmuch  as  the  corresponding  value  of  v  [viz.,  the 
quadrant]  has  been  taken  as  unity  in  the  table  of  numerical 

values  of  jdv  cos  qv2  and  /  dv  sin  qv2. 
If  in  the  formula 

V        2a~ 

we  substitute  for  n  the  value  corresponding  to  a  minimum  of 
the  first  order — i.  e.,  to  the  darkest  part  of  the  first  dark  band, 
we  have 

z=1.873 


61.  If,  however,  we  assume  that  the  fringes  are  produced 
by  the  meeting  of  the  direct  rays  with  those  reflected  at  the 
edge  of  the  opaque  screen,  and  if  we  suppose  further  that  the 
reflected  rays  lose  half  a  wave-length,  we  have  [section  20]  for 
the  same  band 


or   x= 


Accordingly,  these  two  quantities  are  in  the  ratio  of  2  to  1.873. 
The  second  is  measurably  smaller  than  the  first,  differing  as 
they  do  by  nearly  a  fifteenth;  so  that  by  accurate  observations 
on  homogeneous  light  of  well-determined  wave-length  one 
might  distinguish  between  these  two  theories  by  means  of  ex- 
periment. 

62.  The  method  which  I  at  first  thought  best  adapted  to 
the  determination  of  the  wave-length  was  to  measure  the  size 
of  the  fringes  produced  by  two  mirrors  slightly  inclined  to  one 
another,  and  also  the  distance  between  the  two  images  of  the 
luminous  point;  but  the  slightest  curvature  in  the  mirrors 
diminishes  the  accuracy,  and  so  I  preferred  to  use  the  bands 

*  [The  general  expression  for  this  difference  of  path,  d,  has  been  given 
above  —  section  20. 

ax* 
~2b(a+b)' 

Put  d=£,  and  we  haw  the  value  of  x  in  question.] 

127 


MEMOIRS    ON 


produced  by  a  narrow  slit  combined  with  the  cylindrical  lens 
of  which  I  have  already  spoken.  We  have  already  found  that 
the  distance  between  any  two  consecutive  dark  bands,  either  to 

the  right  or  the  left  of  the  aperture,  is  — ,  where  X  is  the  wave- 

c 

length,  c  the  width  of  the  aperture,  and  b  its  distance  from  the 
micrometer.  The  distance  between  the  two  dark  bands  of 
the  first  order  is  just  twice  this  amount.  With  these  data,  it 
is  an  easy  matter  to  determine  X  from  measurements  on  the 
fringes. 

The  following  table  gives  the  results  of  five  observations  of 
this  kind,  together  with  the  wave-lengths  computed  from  them. 
In  order  to  describe  all  the  conditions  of  the  experiment,  I 
include  in  the  table  the  various  values  of  #,  the  distance  from 
the  luminous  point  to  the  screen,  even  though  this  quantity  is 
not  employed  in  the  calculation.  These  measures  have  been 
made  with  practically  homogeneous  red  light,  obtained  by  use 
of  the  same  colored  glass  which,  for  the  purpose  of  getting 
results  that  are  comparable,  I  have  used  in  all  my  observa- 
tions. Each  measure  recorded  in  the  table  is  the  mean  of  four 
observations. 


Number  of 

Distance  from 
luminous  point 
to  diaphragm 

Distance  from 
diaphragm  to 
micrometer 

Size  of 
aperture 

bands  —  in- 
c 
eluded  in  each 

Mean  of 
micrometer 
measures 

Wave  lengths 
computed  from 
these  measurer 

measure 

a 

b 

m. 

m. 

mm. 

mm. 

mm. 

2.507 

1.140 

2.00 

6 

2.185 

0.000639 

2.010 

1.302 

4.00 

10 

2.075 

0.000637 

2.010 

1.302 

3.00 

8 

2.222 

0.000640 

1.304 

2.046 

3.00 

8 

3.466 

0.000635 

1.304 

2.046 

2.00 

6 

3.922 

0.000639 

Sum  of  these  results^          0.003190 

Fifth  of  the  sum  or  mean—  0.000638 

The  agreement  of  these  results  with  each  other  is  very  satis- 
factory, differing,  as  they  do,  among  themselves  by  less  than  one 
per  cent.  Accordingly  I  have  adopted  the  value  0.000.638mm., 
and  have  employed  it  in  all  my  comparisons  of  theory  and  ex- 
periment. 

[Four  pages  devoted  to  verifications  of  the  results  in  this  table 
are  here  omitted.] 

128 


THE    WAVE-THEORY    OF    LIGHT 

65.  Having  thus,  by  means  of  simple  and  well-known  meth- 
ods, verified  the  wave-length  determination  made  with  the 
single  slit  and  cylindrical  lens,  I  have  used  this  same  value  to 
compute  the  exterior  fringes  by  use  of  the  formula 


S  =  IH>  +  *)*X, 

2a 

substituting  for  w  those  values  of  v  which,  according  to  the 
table,  give  maxima  and  minima. 

The  table  on  page  130  summarizes  the  results  of  calculation 
and  observation.  In  my  experiments  I  have  measured  the 
positions  of  the  minima  only,  because  I  considered  this  a  suf- 
ficient test  of  the  theory,  and  because  my  eye  can  determine  the 
darkest  point  of  a  dark  band  with  greater  accuracy  than  it  can 
set  upon  the  brightest  point  of  a  bright  baud. 

[Only  every  fifth  observation  in  the  table  is  reproduced.] 

More  striking  agreement  between  theory  and  experiment 
could  scarcely  be  expected.  W^hen  these  small  differences  are 
compared  with  the  quantities  measured,  and  when  the  great 
variations  in  the  quantities  a  and  b  are  noted,  one  can  no  longer 
doubt  that  the  integrals  which  led  to  these  Jesuits  accurately 
describe  the  law  governing  the  phenomena.  The  probability 
in  favor  of  the  new  theory  is  still  further  increased  by  the  fact 
that  the  wave-length  here  employed  has  been  deduced  from 
different  and  simpler  phenomena. 

[Four  pages,  devoted  to  a  description  of  some  experimental 
precautions  and  to  a  computation  of  exterior  fringes  on  Young's 
hypothesis  of  reflection  from  the  edge  of  the  opaque  body,  are 
here  omitted.} 

68.  We  have  just  seen  that  both  the  formation  and  the  posi- 
tion of  the  exterior  fringes  can  be  explained  in  a  satisfactory 
manner  by  considering  them  as  produced  by  the  meeting  of  an 
infinitely  great  number  of  secondary  waves  which  originate  on 
that  part  of  the  primary  wave  which  is  not  intercepted  by  the 
opaque  screen.  From  this  view  it  follows  that  the  light  which 
is  inflected  into  the  shadow  ought  not  to  produce  any  bright  or 
dark  band,  but  ought  to  diminish  gradually  in  intensity,  pro- 
vided the  screen  is  sufficiently  large  to  allow  no  light  to  go 
around  the  other  side;  and  this  is  true,  even  though  this 
inflected  light,  like  that  which  gives  rise  to  the  exterior  fringes, 
is  the  resultant  of  an  infinitude  of  secondary  waves.  This  will 
i  129 


MEMOIRS    ON 


TABLE    COMPARING    THE    RESULTS    OF    EXPERIMENT    WITH 

THOSE     OF    THEORY 
EXTERIOR    FRINGES   IN    RED   LIGHT   OP   WAVE-LENGTH   0.000638  MM. 


Number 
of  obser- 
vation 

Distance 
from  lumi- 
nous point 

to  0]),'lc|llf 

screen 
a 

Distance  from 
op;.  quo  body 
to  microme- 
ter 

/) 

Order  of 
dark 
band 

Distil  noe  from  darkest 
point  in  each  band  to 
edge  of  geometrical 
shadow 

Difference 

Observed 

Computed 

in. 

m. 

mm. 

mm. 

ri 

2.84 

283 

-1 

3 

414 

4  14 

0 

1 

0.1000 

07985 

is 

5.14 

5.13 

-1 

14 

5.96 

5.96 

0 

15 

6.68 

6.68 

0 

fl 

1.05 

1  05 

0 

N 

1.54 

1  54 

0 

5 

0.510 

0.501 

i  3 

1.90 

1.91 

+1 

4 

2.21 

222 

+1 

15 

2.49 

2.49 

0 

fl 

2.59 

259 

0 

2 

3.79 

3.79 

0 

10 

1.011 

2.010 

^3 

4.68 

4.69 

+1 

14 

5.45 

5.45 

0 

15 

6.10 

6.11 

+1 

fl 

0.54 

0.55 

+1 

2 

0.80 

0.81 

+1 

15 

3.018 

0.253 

I3 

1.00 

1.00 

0 

I4 

1.16 

1.16 

0 

15 

1.31 

1.31 

0 

fi 

3.19 

3.22 

+  3 

2 

4.70 

4.71 

+  1 

20 

3.018 

3995 

\z 

5.83 

5.84 

+  1 

4 

6.73 

6.78 

+  5 

U 

7.58 

7.60 

+  2 

fl 

1.13 

1.14 

+  1 

2 

1.67 

.1.67 

0 

25 

6.007 

0.999 

<S 

2.06 

2.07 

+  1 

I4 

2.40 

2.49 

0 

16 

2.69 

2.69 

0 

be  easily  seen  by  looking  at  the  table  below,  which  gives  the 
intensity  of  light  in  the  shadow  for  rays  inflected  at  various 
angles.  These  intensities  have  been  computed  by  means  of  the 
table  giving  the  numerical  values  of  the  integrals 

/  dv  cos  qv2  and  /  dv  sin  qv2, 
130 


THE    WAVE-THEORY    OF    LIGHT 


by  taking  the  sums  of  the  squares  of  the  corresponding  numbers 
and  subtracting  ^.  In  spite  of  the  inaccuracy  introduced  by 
the  method  of  partial  integration  employed  in  the  first  table, 
it  is  seen  that  the  intensity  of  light  diminishes  rapidly  as  v 
increases,  presenting  none  of  the  maxima  and  minima  observed 
outside  of  the  shadow. 

INTENSITIES     OF     LIGHT     DIFFRACTED     INTO     THE     SHADOW 
UNDER    DIFFERENT     ANGLES 


Values  of  v 

Corresponding 
intensities 

Values  of  v 

Corresponding 
intensities 

q 
0.10 

04095 

q 
2.90 

0.0121 

020 

03359 

3.00 

0.0113 

030 

0.2765 

3.10 

0.0105 

0.40 

0.2284 

3.20 

0.0098 

0.50 

0.1898 

3.30 

00092 

060 

0.1586 

3.40 

0.0087 

0.70 

0.1334 

3.50 

0.0083 

0.80 

0.1129 

3.60 

0.0079 

0.90 

0.0962 

3.70 

0.0074 

1.00 

0.0825 

3.80 

00069 

1.10 

00711 

3.90 

0.0066 

1  20 

0.0618 

4.00 

0.0064 

1  30 

0.0540 

4.10 

0.0061 

1.40 

0.0474 

4.20 

0.0057 

1.50 

0.0418 

430 

0.0054 

1.60 

0.0372 

4.40 

0.0052 

1.70 

00332 

4.50 

0.0051 

1.80 

0.0299 

4.60 

00048 

1  90 

0.0271 

4.70 

0.0045 

2.00 

0.0247 

4.80 

00044 

210 

0.0226 

4.90 

0.0043 

2.20 

0.0207 

5.00 

0.0041 

2.30 

0.0189 

5  10 

00038 

2.40 

0.0173 

520 

00037 

2.50 

00159 

5.30 

0.0036 

260 

0.0147 

5.40 

0.0035 

2.70 

0.0137 

5.50 

0.0033 

2.80 

0.0129 

As  usual,  a  and  b  represent  the  distances  of  the  screen  from 
the  luminous  point  and  from  the  plane  in  which  the  shadow  lies, 
while  x  is  the  distance  from  the  edge  of  the  geometrical  shadow 
to  the  point  in  this  plane  under  consideration,  so  that  we  have 


131 


MEMOIRS    ON 
and  therefore 


69.  By  the  aid  of  these  formulae  we  can  find  the  value  of  the 
distance  x  or  the  angle  x/b  of  the  inflected  ray  corresponding 
to  the  various  values  of  #;  and  vice  versa,  if  x  or  the  slant  x/b 
be  given,  we  can  find  v,  and  thus  determine  the  intensity  of  the 
inflected   light.     One   striking   inference   from  this  formula, 

\a~\    ' — ,  is  that  the  values  of  x  are  not  directly  pro- 
6(1 

portional  to  those  of  b,  but  are  related  to  them  as  the  ordinates 
of  a  hyperbola  are  to  its  abscissas.  It  thus  follows  that  points 
of  equal  intensity  along  the  edge  of  the  geometrical  shadow  do 
not  lie  upon  a  straight  line  as  we  vary  b,  but  upon  a  hyperbola 
of  appreciable  curvature,  like  the  corresponding  loci  in  exterior 
fringes. 

70.  I  have  not  yet  succeeded  in  verifying  by  direct  experi- 
ment the  ratios  of  intensity  in  the  inflected  light  as  predicted 
by  the  theory  of  interference  applied  to  the  principle  of  Huy- 
gens.     A  measurement  of  this  kind  is  very  difficult  [foot-note 
omitted],  and  I  hardly  think  that  one  would  be  able  to  reach 
the  same  accuracy  as  in  the  determination  of  the  darkest  and 
brightest  points  in  fringes.     The  results  already  obtained  for 
fringes,  however,  appear  to  me   as  verifications — indirect,  it 
must   be   confessed  —  of   these  very  ratios   of  intensity;    for 
whenever  the  positions  of  maxima  and  minima  have  been  de- 
duced from  the  general  expression  for  the  intensity  of  light 
and  have  been  found  to  coincide  accurately  with  experiment, 
it   becomes  more  and   more  probable  that  this  integral  cor- 
rectly represents  all  the  variations  of  intensity  in  the  inflected 
light. 

71.  In  the  case  of  exterior  fringes  one  may,  as  we  have  seen, 
use  the  table  of  maxima  and  minima  to  compute  the  positions 
of  the  darkest  and  brightest  points  in  the  dark  and  bright  bands 
for  all  values  of  a  and  b.     This,  however,  is  not  the  case  with 
regard  to  the  interior  fringes  in  the  shadow  of  a  narrow  body 
or  in  the  case  of  a  narrow  aperture.    The  limits  of  the  integral 
vary  all  the  while,  and  it  is  therefore  impossible  to  give  general 
results  applicable  to  every  case,  so  that  one  is  obliged  to  deter- 
mine the  maxima  and  minima  for  each  particular  case,  using 

132 


THE    WAVE-THEORY    OF    LIGHT 


the  table,  which  gives  the  numerical  values  of  j  dv  cos  qv2  and 

/  dv  sin  qv2.  I  propose  to  give  the  results  of  all  the  computa- 
tions of  this  kind  which,  up  to  the  present,  I  have  made  for  the 
purpose  of  testing  the  theory.  They  are  very  long,  and  I  have 
not  been  able  to  finish  as  many  as  I  had  desired,  but  this  lack  in 
quantity  is,  perhaps,  compensated  by  the  variety  of  the  cases 
which  I  have  studied,  for  in  trying  the  theory  on  the  observa- 
tions, I  have,  by  preference,  selected  cases  in  which  the  disposi- 
tion of  the  fringes  is  somewhat  unusual. 

72.  And,  first,  I  propose  to  consider  the  case  of  a  narrow 
aperture  which  presents  at  once  the 
case  of  exterior  and  interior  fringes. 
Let  C  be  a  luminous  point,  AG 
a  narrow  aperture  whose  edges  A 
and  G  are  straight  and  parallel  ; 
let  BD  be  the  central  projection  of 
this  aperture  upon  the  plane  in 
which  the  fringes  are  observed, 
and  P  be  any  point  in  this  plane 
at  which  the  intensity  is  to  be  de- 
termined. For  this  purpose  we 

must  integrate 

//     g* 
dz  cos  ( 2q 

and 


''(a±b]\ 
ab\     I 


I  dz  sin  l*2q 


ab\    1 

between  the  limits  A  and  G,  afterwards  taking  the  sum  of  the 
squares  of  these  integrals.  This  will  give  us  the  intensity  of 
light  at  the  point  P,  but  we  must  not  forget  that  the  origin 
from  which  z  is  measured  lies  upon  the  direct  line  CP,  and 
that  therefore  the  two  limits  A  and  G  correspond  to  z=MG 
and  z=—  AM.  The  next  step  is  to  compute  the  corresponding 
values  of  v  from  the  formula 


/2 

v=zy- 


or 


133 


MEMOIRS    ON 

where  x  is  the  distance  from  the  point  P  to   the  edge  of  the 
geometrical  shadow.     From  the  table  of  integrals, 

/  dv  cos  qv2  and    /  dv  sin  qv2, 

we  then  find  the  values  most  nearly  corresponding  to  those  of  v. 
Let  us  call  t  the  difference  between  the  value  for  which  the 
integral  is  desired  and  the. number  i  [for  which  it  is  computed] 
in  the  table.  The  proper  integral  can  then  be  found  by  means 
of  the  formulas  of  approximation. 

/i+t  /»  i  -| 

dv  cos  qv2=  I  dv  cos  qv*  +  — —(sin  qi  (i'-\-*Zt)—  sin  qi2) 
Jo  2iq 

/i+t  rl  i 

dv  sin  qv2=.J  dv  sin  qv2 -f- T-T-  (—cos  qi  (i  +  '2t)-{-cos  qi2). 

Having  made  this  computation  for  the  two  values  of  v  which 
represent  the  edges  A  and  G-  of  the  aperture,  if  the  point  is  in- 
side we  add  these  integrals;  if,  however,  it  is  on  the  outside, 
we  subtract  them  ;  and,  lastly,  take  the  sum  of  the  squares  of 
the  two  numbers  thus  found.  In  like  manner,  one  finds  the 
intensity  of  the  light  for  any  other  point  whose  position  is 
given,  and  in  comparing  these  various  results  the  positions  of 
maxima  and  minima  may  be  found. 

[Half  a  page  concerning  the  method  of  interpolation  is  here 
omitted.] 

73.  In  order  to  apply  this  method  of  computation  to  the  ob- 
servations, I  first  determined  the  tabulary  value  of  c,  that  is  to 
say,  the  size  of  the  aperture,  by  means  of  the  formula 


f,= 


/2(a 
V 


so  that  I  thus  obtained  the  tabular  interval  between  the  limits. 
By  a  few  easy  trials  I  find  between  what  numbers  of  the  table 
the  maxima  and  minima  lie  ;  afterwards  I  determine  their  posi- 
tion more  accurately  by  the  process  which  I  have  just  de- 
scribed. Having  thus  obtained  the  values  of  v  corresponding  to 
maxima  or  minima,  I  subtract  them  from  the  half  of  the  tabulary 
value  of  c,  in  order  to  refer  them  to  the  middle  of  the  aperture. 
And,  last  of  all,  the  for  inn  hi 


2a 

gives  me  the  distance  of  these  same  maxima  or  minima  from 
the  middle  of  the  projection  of  aperture,  which  is  the  point  of 
reference  used  in  my  observations. 

134 


THE    WAVE-THEORY    OF    LIGHT 


COMPARISON    OP   THEORY    AND    EXPERIMENT 

REGARDING    THE    POSITIONS    OF    MAXIMA    AND    MINIMA     IN    THE    FRINGES 
PRODUCED   BY    A    NARROW    APERTURE 


Number     of    bright 
or     dark       bands 
counted  from  mid- 
dle 

Approximate 
value  of* 
counted  from 
edge  of  aper- 
ture 

Corresponding 

intensity 

Value  of*  cor- 
responding to 
itftxitH't  or 
minim*  i 

Distance   of   maxima 
or     minima      from 
projection  of  centre 
of  aperture 

Difference 

Computed 

Observed 

FIRST  OBSERVATION 

m.                      m.               mm. 

a=2.QW  ;  6=0.617;  c=0.50  ;  tabulary  value  of  e=1.288 

mm. 

mm. 

mm. 

(  +0.812 

0.03495  ) 

1.  Minimum 

4  4-0.912 

0.01645  L 

+0.913 

0.79 

0.77 

+0.02 

(  +1.012 

0.03406  )  . 

(  +2.412 

0.00238  ) 

2.  Minimum 

]  +2.512 

0.00235  }• 

+2.463 

1.58 

1.58 

0.00 

(  +2.612 

0.00541  ) 

THIRD   OBSERVATION 

m.                    m.                 mm. 

a=2.010;  6=0.401;  c  =  1.00;  tabulary  value  of  c=3.062 

mm. 

mm. 

mm. 

(  -1.262 

2.2575  ) 

1.  Minimum 

-   -1.162 

2.2153  [ 

-1.181 

0.14 

0.16 

-0.02 

(  -1.100 

2.2577  ) 

(  -0.300 

0.7135  ) 

2.  Minimum 

-0.262 

0.6925  [• 

-0.215 

0.51 

0.48 

+0.03 

(  -0.162 

0.6950  ) 

(  +0.400 

0.1501  ) 

3.  Minimum 

•<  +0.438 

0.1477V 

+0.431 

0.77 

0.76 

+0.01 

(  +0.500 

0.1604  ) 

(  +0.938 

0.0799  ) 

4.  Minimum 

1  +1.038 

0.0417V 

+1.084 

1.02 

1.01 

+  .01 

(  +1.138 

0.0432  J 

(  +1.800 

0.0170  ) 

5    Minimum 

\  +1.738 

0.0128  [ 

+  1.736 

1.28 

1.28 

0 

(  +1.700 

0  0141  ) 

FIFTH   OBSERVATION 

m.                     m.               mm. 

«=2.010;  6=0.492;  c  =  1.50;  tabulary  value  of  c=  4.  224 

mm. 

mm. 

mm. 

(  -1.300 

2.7239  ) 

1.  Maximum 

-  -1.200 

3.0466  - 

i  -1.168 

042 

0.43 

-0.01 

/  -1.100       2.9780)  A 

[The  second,  fourth,  and  sixm  observations  are  omitted.] 
135 


CA,  ,,-, 


MEMOIRS    ON     . 

Evidently  theory  and  observations  agree  in  general  quite 
well,  although  in  the  second  and  fourth  observations  the  dis- 
agreement is  quite  marked  and  rather  more  than  one  would 
expect  from  the  size  of  the  fringes  ;  for  in  the  second  observa- 
tion the  individual  measures  differ  at  most  by  0.04  mm.,  and 
the  fourth  observation,  which  I  have  already  described,  agrees 
perfectly,  as  has  been  seen,  with  another  experiment  in  which 
the  same  fringes  appear.  This  disagreement,  therefore,  can 
only  be  explained  by  assuming  that  the  theory  is  wrong  or  that 
constant  errors  have  entered  the  observations  through  optical 
illusion. 

74.  Our  theory  rests  upon  a  hypothesis  which  is  at  once  so 
simple  and  so  inherently  probable,  and  which  besides  has  been 
so  strikingly  verified  by  many  varied  experiments,  that  one  can 
scarcely  doubt  the  truth  of  the  fundamental  principle.  It  is 
quite  possible  that  this  anomaly  is  only  apparent,  and  that  the 
eye  does  not  correctly  estimate  the  position  of  the  minima  in 
question.  We  must  remember  that  they  are  not  very  sharp, 
and  that  they  are  always  bounded  on  each  side  by  two  bright 
bands  of  very  different  intensities.  Now,  in  order  to  deter- 
mine the  position  of  the  minimum,  my  eye  must  include  a  part 
of  each  of  these  two  bands,  so  that  that  part  of  the  dark  band 
on  the  side  next  the  brightest  appears  to  me  darker  still  on 
account  of  its  environment ;  thus  attracting,  as  it'  were,  the 
apparent  minimum  to  its  side,  and,  indeed,  all  the  discrepancies 
lie  in  this  direction.  That  the  eye  includes  a  sufficiently  large 
portion  of  the  fringes  for  correctly  estimating  the  position  of 
maxima  and  minima  is  evident  from  the  fact  that  in  repeating 
the  fourth  observation,  using  a  diaphragm  of  small  aperture  in 
the  focus  of  the  micrometer  eye-piece,  nothing  was  left  but  a 
band  which  was  uniformly  dark  and  in  which  the  minimum 
was  no  longer  distinguishable.  If  I  have  succeeded  in  getting 
the  correct  positions  of  the  minima  in  the  exterior  fringes 
even  in  regions  of  poor  definition,  it  is  owing  to  the  fact  that 
the  bright  bands  between  which  these  are  included  differ  very 
slightly  in  intensity  ;  and  if,  in  the  case  of  the  narrow  aperture 
and  cylindrical  lens,  experiment  and  theory  happen  to  agree  in 
spite  of  great  differences  of  intensity  between  two  adjoining 
bright  bands,  this  is  because  the  dark  band,  especially  in  the 
first  and  second  orders,  is  almost  perfectly  black.  In  general, 
whenever  the  maximum  or  minimum  is  very  sharp,  I  find.ox- 

136 


THE    WAVE-THEORY    OF    LIGHT 


perirnent  and  calculation  in  thorough  agreement.  In  the  fifth 
observation,  for  instance,  I  measure  the  distance  from  the  centre 
to  the  maximum  of  the  first  order  because  this  bright  band  is 
very  well  defined,  and  I  am  therefore  able  to  determine  its  most 
brilliant  point  with  great  precision.  The  difference  between 
the  computed  and  observed  values  is  indeed  only  0.01  mm. 

75.  But  our  theory  does  more  than  merely  give  us  the  posi- 
tions of  maxima  and  minima,  for  it  enables  us  to  predict  the 
general  appearance  of  the  phenomena,  so  that,  without  experi- 
mental determination,  we  can  foretell  the  variations  of  intensity 
in  the  light ;  thus,  for  instance,  in  the  fifth  observation  the  part 
of  the  shadow  corresponding  to  the  middle  of  the  aperture  was 
filled  by  a  large  dark  band  of  a  tint  that  was  practically  uni- 
form up  to  0.26  mm.  on  each  side  of  the  centre,  after  which 
the  intensity  of  the  light  increased  rapidly  so  as  to  form  the 
bright  band  of  the  first  order  which  I  have  just  mentioned. 
Now  in  computing  the  intensity  of  the  light  within  these 
limits,  we  find  that  in  fact  its  intensity  varies  scarcely  at  all, 
but  that  in  passing  from  these  limits  to  the  bright  band  it  in- 
creases very  rapidly.  In  the  following  table  are  given  the  re- 
sults of  computation  for  different  points  of  the  dark  band  and 
the  two  bright  bands  which  include  it.  The  position  of  each 
point  is  denoted  by  the  corresponding  value  of  v,  measured  al- 
ways from  one  of  the  edges  of  the  aperture. 


Number  of 
observation 

Value  ofv 

Intensity 

1 

1.100 

2.9780 

2 

1.200 

30466 

3 

1.300 

2.7239 

4 

1.400 

2.2843 

Limit   of  the  colored  ) 

region  as  determined  > 

5 

1.524 

1.9671 

by  observation.           ) 

6 

1.824 

1.9100 

7 

2.112 

1.9802 

The  distribution  of  intensities  on  the  other  side  of  the  centre  is 

the  same. 

If  the  distances  of  these  various  points  from  a  common  ori- 
gin be  plotted  as  abscissas  and  the  corresponding  intensities  as 
ordinates,  we  shall  obtain  the  curve  MCM',  which  gives  us  in 
fat  t~a  picture  of  the  phenomenon  just  as  one  finds  it  in  the 

137 


MEMOIRS    ON 


M 


J — 1 l__l — I 1 L 

12345  6  7 

*iu.  2 1 


Fiy. 

experiment.  I  should  like  to  have  made  similar  drawings  for 
all  the  other  observations  in  order  to  facilitate  the  comparison 
of  theory  with  experiment,  but  the  length  of  the  computation 
and  the  time  at  my  disposal  did  not  permit. 

[Five  pages,  in  which  the  case  of  a  narrow  opaque  obstacle  is 
discussed,  are  here  omitted.] 

79.  I  have  now  applied  the  principle  of  Huygens  to  the 
three  general  classes  of  phenomena  in  which  diffraction  occurs, 
namely,  first,  to  the  fringes  produced  by  a  screen  whose  edges 
are  straight  and  infinitely  long,  and  which  is  so  large  that  the 
light  passes  practically  only  one  edge  of  the  screen  ;  secondly, 
to  the  fringes  produced  by  a  system  of  two  similar  screens 
brought  very  near  together  ;   thirdly,  to  those  fringes  which 
accompany  the  shadow  of  a  very  narrow  screen.* 

Comparing  observations  with  the  predictions  of  the  theory, 
I  have  shown  that  it  suffices  to  explain  the  most  diverse  phe- 
nomena, and  that  the  general  expression  for  the  intensity  of 
light  derived  from  it  gives  us  a  faithful  picture  of  the  phenom- 
ena, even  when  they  are  most  bizarre  and  apparently  irregular, 

Besides  the  three  general  classes,  one  might  devise  a  large 
number  of  others  by  combining  these  among  themselves.  The 
theory  would  doubtless  apply  here  with  the  same  success  and 
the  same  ease.  The  computation  would  be  more  tedious  in 
proportion  as  the  variety  of  limits  assigned  to  the  integrals  be- 
came greater  and  greater  ;  the  experiments  would  also  demand 
more  complicated  apparatus. 

80.  In  the  first  section  of  this  memoir  I  have  described  a 
phenomenon  which  results  from  a  combination  of  two  of  the 
principal  cases  of  diffraction,  namely,  the  fringes  produced  by 

*  I  do  riot  here  include  those  fringes  which  are  produced  by  the  biprism, 
or  two  mirrors  slightly  inclined  to  each  other,  for,  properly  speaking, 
these  are  not  diffraction  effects,  since  they  are  not  produced  by  rays  which 
are  diffracted  or  inflected,  but  by  two  pencils  which  are  regularly  re 
fleeted  or  refracted. 

138 


THE    WAVE-THEORY    OF    LIGHT 


light  in  passing  through  two  apertures,  each  very  narrow  and 
each  near  to  the  other.  Having  prepared  a  sheet  of  copper  in 
the  form  drawn  in  Fig.  15,  I  noted  that 
when  the  large  fringes  produced  by  each 
of  the  slits  CEC'E'  and  DFD'F',  expand- 
ing as  I  moved  away  from  the  screen,  had  G_ 
filled  the  shadow  6f  CDFE  so  that  it  con-  " 
tained  only  the  bright  band  of  the  first 
order,  the  interference  bands  resulting 
from  the  twc  pencils  of  light  became 
much  sharper  and  brighter  than  the  in- 
terior fringes  of  the  part  ABCD.  The 


Fig.  15 


lower  part,  CEDF,  which  was  at  first  brighter  than  the  other, 
became  darker  the  farther  I  went  away  from  the  screen,  but 
its  fringes  continued  to  show  colors  which  in  white  light  were 
purer  and  bands  which  in  homogenous  light  were  sharper. 
With  the  simple  apparatus  which  I  employed  one  could  not  ob- 
tain exact  measures,  and  I  have  not  therefore  carried  out  the 
computations  for  this  experiment ;  accordingly  I  limit  myself 
to  the  explanation  of  these  phenomena  by  means  of  some  gen- 
eral considerations. 

Let  L  be  the  luminous  point,  and  IK  the  horizontal  projec- 
tion of  the  part  AEBF  of  the  screen  represented  in  Fig  15.  P  is 
any  point  in  the  interior  of  the  shadow  lying  upon  the  straight 
line  LO.  From  the  point  L  as  centre,  and 
with  a  radius  equal  to  LI,  describe  the  arc 
IMM',  representing  the  incident  wave.  Now 
with  a  point  P  as  centre,  arid  with  a  radius 
equal  to  IP,  describe  the  arc  Imm'.  The  vari- 
ous distances  between  these  two  arcs  give  us 
the  differences  of  path  traversed  by  the  sec- 
ondary waves  meeting  at  P.  We  shall  first 
consider  the  upper  part  of  the  screen — that  is, 
the  case  where  the  wave  IMM'  is  not  inter- 
cepted on  the  other  side  of  the  point  I.  Let 
us  now  imagine  this  wave  divided  into  a 
large  number  of  small  arcs,  IM,  MM',  etc., 
in  such  a  manner  that  the  straight  lines 
drawn  to  P  from  any  two  consecutive  points 
of  division  differ  by  half  a  wave-length ;  and, 
for  sake  of  simplicity,  let  us  suppose  that  the 
139 


MEMOIRS    ON 

point  P  lies  well  within  the  edge  of  the  shadow,  or,  what  is  the 
same  thing,  let  us  imagine  the  ray  IP  sufficiently  inclined  to 
the  incident  ray  to  make  these  arcs  practically  equal.  Then 
each  of  these  arcs,  excepting  the  one  at  the  end  IM,  will  lie  be- 
tween two  others,  which  will  combine  to  annul  its  effect  at  the 
point  P.  In  the  case  of  the  arc  IM,  which  lies  at  the  extreme 
edge  of  the  wave,  we  have,  however,  an  exception  ;  for  this  arc 
loses  only  one-half  its  intensity  by  interference  with  the  vibra- 
tions of  the  neighboring  arc,  MM'.  If,  therefore,  we  intercept 
this  arc  [MM']  and  all  the  rest  of  the  incident  wave,  the  light 
which  is  received  by  the  point  P  will  actually  be  increased  ;* 
this  is  precisely  the  effect  which,  at  a  certain  distance,  is  pro- 
duced by  the  part  of  the  screen  G'C'E"  (Fig.  15).  But  in 
proportion  as  the  point  P  (Fig.  22)  recedes  from  the  opaque- 
screen,  the  arc  \mrti  approaches  the  wave  IMM' ;  and  in  the 
case  where  the  luminous  point  L  is  at  an  infinite  distance, 
these  two  approach  indefinitely  near  to  each  other.  The  di- 
visions M,  M',  etc.,  being  determined  by  the  separation  of 
these  two  arcs,  keep  spreading  apart  from  the  point  I  in  pro- 
portion as  the  arcs  approach  each  other.  It  follows,  therefore, 
that  the  part  MI  of  the  incident  wave  will  grow  larger  and 
larger,  and  the  rays  from  this  part  passing  the  edge  0  (Fig.  15) 
retain  at  least  half  their  intensity  in  the  region  behind  the 
upper  part  of  the  screen.  But  in  the  lower  part  of  the  screen 
the  aperture  CEC'E'  does  not  increase  in  size,  so  that  if  the 
luminous' point  is  far  enough  away  the  effective  arc  IM  (Fig.  22) 
will  finally  become  so  large  compared  with  this  aperture  that 
[most  of  the  rays  from  MI  are  intercepted  by  GC'E',  and  hence] 
the  point  will  receive  less  light  in  the  lower  part  of  the  shadow 
than  in  the  upper. 

Let  us  now  pass  to  the  consideration  of  fringes  produced  by 
the  meeting  of  rays  coming  from  both  edges  of  the  screen, 
AEBF  (Fig.  15).  Behind  the  upper  part,  ABOD,  the  inflected 
light  diminishes  rapidly  in  intensity  .as  one  recedes  from  the 
edge  of  the  geometrical  shadow,  and  therefore  all  the  fringes 
except  those  which  are  very  near  the  middle  are  produced  bj 
two  rays  of  very  unequal  intensities ;  consequently  the  dark 

*  The  light  at  P  would  be  increased  still  more  if  the  screen  were  perfo- 
rated in  such  a  way  as  to  permit  all  the  arcs  of  even  order  to  pass  through 
and  at  the  same  time  intercept  all  the  arcs  of  odd  order. 

140 


THE    WAVE-THEORY    OF    LIGHT 

bands  are  not  very  sharp  when  one  uses  homogeneous  light, 
and  the  colors  are  mixed  with  gray  when  one  uses  white  light. 
Behind  the  lower  part,  CEDF,  the  two  pencils  of  light  coming 
from  the  slits  CEO'E  '  and  DFD'F '  have  a  practically  uniform 
intensity  throughout  a  considerable  portion  of  the  bright  band 
from  each  of  these  apertures  ;  and  if  these  apertures  are  so  nar- 
row compared  with  the  distance  between  them  that  the  region 
of  uniform  intensity  in  the  inflected  light  includes  all  the 
fringes  produced  by  the  two  pencils,  then  in  those  points  where 
the  vibrations  are  in  complete  discordance  the  light-waves  will 
almost  completely  destroy  one  another  ;  accordingly  the  dark 
bands  will  be  very  much  sharper  than  in  the  upper  part  of  the 
shadow  when  homogeneous  light  is  employed,  and  the  colors 
will  be  very  much  purer  when  white  light  is  used.  When  one 
looks  at  these  points  close  up  to  the  screen  before  the  larger 
fringes  which  are  produced  by  each  slit  have  spread  out  into 
the  shadow  AEBF,  the  phenomenon  becomes  very  complicated 
and  changes  rapidly  with  the  distance  of  the  rnagnifying-glass, 
especially  when  the  distance  between  the  two  slits  is  not  very 
great  when  compared  with  their  size.  It  would  be  interesting 
to  determine  by  computation  the  positions  of  the  maxima  and 
minima  of  the  bright  and  dark  bands,  and  to  compare  these  re- 
sults with  those  of  observation.  I  have  no  doubt  that  the  the- 
ory would  thus  acquire  fresh  confirmation. 

81.  Hitherto  we  have  considered  all  waves  as  coming  from  a 
single  centre,  but,  in  actual  experiment,  luminous  points  are 
always  made  up  of  a  very  large  number  of  centres  of  vibration, 
and  it  is  to  each  one  of  these  by  itself  that  the  preceding  dis- 
cussion applies.  So  long  as  these  are  not  very  widely  sepa- 
rated from  each  other,  the  fringes  which  they  produce  practi- 
cally coincide,  but  the  dark  bands  from  one  overlap  the  bright 
bands  of  the  other  in  proportion  as  we  increase  the  dimensions 
of  the  luminous  point,  until  finally  they  completely  annul  each 
other.  In  the  case  of  the  exterior  fringes  this  effect  is  more 
and  more  appreciable  as  one  gets  farther  and  farther  away 
from  the  screen,  because  it  increases  directly  as  the  distance, 
while  the  size  of  the  bright  and  dark  bands  increases  less  rapid- 
ly. And  this  is  why  a  luminous  source  sufficiently  small  to 
produce  fringes  which,  in  the  near  neighborhood  of  an  opaque 
body,  are  very  sharp  will,  at  a  considerable  distance  from  this 
body,  give  only  ill-defined  fringes. 

141 


MEMOIRS    ON 

82.  It  is  not  necessary  that  the  interposed  body  should  be 
opaque  in  order  to  produce  the  phenomena  of  diffraction  at  its 
edges  ;  all  that  is  required  is  that  a  part  of  the  wave  should  be 
retarded  with  respect  to  its  neighboring  parts,  but  this  is  ex- 
actly what  a  transparent  body  does  when  its  refractive  index 
differs  appreciably  from  that  of  the  medium  surrounding  it  ;  it 
thus  gives  rise  to  fringes  which  border  both  the  inside  and  the 
outside  of  their  shadow.     They  are  exactly  like  the  exterior 
fringes  of  opaque  bodies  when  the  difference  of  path  between 
the  rays  which  have  traversed  the  transparent  screen  and  the 
outside  rays  contains  a  considerable  number  of  wave-lengths,  be- 
cause their  mutual  influence  [interference]  is  no  longer  appre- 
ciable and   we  have  simply  the  addition  of  two  uniform  illu- 
minations.   But  this  is  not  the  case  when  the  transparent  screen 
is  very  even  or  when  its  refractive  index  differs  very  slightly 
from  that  of  the  surrounding  medium,  for  now  the  fringes  are 
altered  in  a  very  marked  way  by  the  mutual  influence  of  those 
rays  which  traverse  the  transparent  plate  and  those  which  pass 
its  edge.     It  is  from  similar  reasons  that  the  striae  in  layers  of 
mica  resulting  from  slight  differences  of  thicknesses  give  rise, 
in  white  light,  to  colored  fringes  in  the  very  peculiar  manner 
described  by  M.  Arago. 

83.  As  to  fringes  of  the  kind  which  we  have  called  interior, 
they  are  not  to  be  obtained  with  a  narrow  transparent  body, 
because  the  direct  light  which  traverses  it  is  so  much  brighter 
than  the  inflected  rays  as  to  mask  the^effects  of  interference; 
and,  besides,  the  bright  and  dark  bands  which  this  transparent 
body  tends  to  produce,  when  considered  as  a  narrow  aperture, 
do  not  coincide  with  those  which  it  tends  to  produce  when 
considered  as  a  small  obstacle. 

84.  The  phenomena  of  diffraction,  once  explained  for  the 
case  of  homogeneous  light,  are  easily  predicted  for  the  case  of 
white  light.     These  fringes  come  from  the  superposition  of  all 
the  bright  and  dark  bands  of  the  various  sizes  produced  by  the 
different  kinds  of  waves  which  go  to  make  up  white  light,  so 
that  when  we  have  once  computed  the  intensity  of  each  of  the 
principal  kinds  of  rays  at  the  point  under  consideration,  using 
the  proper  wave-length,  according  to  the  theory  which  I  have 
just  explained,  we  can  find  the  resultant  tint  by  substituting 
these  values  in  Newton's  empirical  formula  for  determining  the 
result  obtained  by  mixing  any  set  of  colored  rays. 

142 


THE    WAVE-THEORY    OF    LIGHT 

85.  Polished  surfaces  illuminated  by  a  point-source  present 
a  set  of  diffraction  phenomena  exactly  like  those  which  we  ob- 
serve in  direct  light.     The  field  of  light  reflected  by  a  mirror 
is  bordered  with  fringes  similar  to  those  which  surround  the 
shadows  of  bodies.     If  the  surfaces  be  very  narrow  or  so  black- 
ened that  only  a  single  bright  line  remains,  or  indeed  if  one 
inclines  the  mirror  in  such  a  way  as  to  diminish  greatly  the 
size  of  the  field  [foot-note  here  omitted],  the  phenomenon  of  a 
pencil  of  light  dilated  by  passing  through  a  very  narrow  aperture 
will  be  reproduced.     If  a  mirror  be  blackened  throughout  its 
entire  extent,  with  the  exception  of  two  bright  lines,  it  gives 
rise  to  a  set  of  fringes  identical  with  those  produced  by  two 
parallel  slits  in  an  opaque  screen.    If,  instead  of  blackening  the 
large  part  of  the  reflecting  surface,  one,  on  the  contrary,  mere- 
ly traces  a  single  fine  black  line,  it  will  produce  fringes  similar 
to  those  observed  in  the  shadow  of  a  narrow  screen.     In  short, 
the  phenomena  are  absolutely  the  same  as  if  the  mirror  were 
transparent  and  the  rays  came  from  the  image  of  the  luminous 
point.     The  explanation  is  very  simple  ;  for  we  know  that  the 
image  (which  lies  upon  the  perpendicular  drawn  from  the  lu- 
minous point  to  the  mirror,  and  which  is  situated  at  a  distance 
from  the  surface  of  the  mirror  equal  to  the  distance  of  the  lu- 
minous point  from  the  mirror),  has  this  remarkable  property, 
namely,  that  its  distance  from  any  point  on  the  surface  of  the 
mirror  is  equal  to  the  distance  of  the  same  point  from  the  lu- 
minous centre.     When,  therefore,  we  consider  the  rays  as  orig- 
inating in  the  image  of  a  luminous  point,  we  do  not  alter  the 
difference  of  path  traversed  by  the  elementary  waves  which 
produce  the  fringes,  and  consequently  there  is  no  change  in 
the  size,  or  in  the  relative  intensities,  of  the  bright  and  dark 
bands. 

I  may  here  remark  that  the  position  [pliase]  of  the  resultant 
of  the  secondary  waves  at  any  point,  depending  as  it  does  merely 
upon  differences  of  path,  ought,  in  the  case  of  reflection,  to  be 
the  same  as  if  the  rays  were  emitted  by  the  image  just  men- 
tioned. Consequently,  in  the  case  of  a  polished  surface  of 
large  area,  all  the  partial  resultants  will  be  situated  at  the 
same  distance  from  this  point,  thus  making  it  the  centre  of 
the  reflected  wave. 

86.  It  is  by  means  of  these  secondary  waves  that  Huygens 
has  explained  in  such  a  simple  manner  the  laws  of  reflection 

143 


MEMOIRS    ON    THE   WAVE-THEORY    OF    LIGHT 

and  refraction,  showing  that  they  are  phenomena  of  the  same 
kind  as  the  propagation  of  light  in  a  homogeneous  medium; 
but  his  explanation  leaves  much  to  be  desired.  He  has  not 
proved  that  there  will  be  only  one  system  of  waves  resulting 
from  this  multitude  of  systems  of  secondary  waves,  for  he  has 
not  used  the  principle  of  interference.  He  assumes  that  the 
light  is  appreciable  only  in  those  points  where  the  secondary 
waves  coincide  [in  phase]  exactly  ;  while  the  complete  absence 
of  any  luminous  disturbance  can  occur  only  when  the  second- 
ary disturbances  are  in  [direct]  opposition.  It  was  this,  doubt- 
less, that  led  him  to  think  that  light  was  not  inflected  to  any 
appreciable  extent  into  shadows,  and  which  prevented  him 
from  discovering  the  phenomena  of  diffraction,  the  laws  of 
which  his  theory  could  have  given  him  without  recourse  to  ex- 
periment. 

This  theory,  when  combined  with  the  principle  of  interfer- 
ence, gives  us  not  only  the  path  of  the  ray  in  the  particular 
case  where  reflection  occurs  at  a  polished  surface  of  indefinite 
extent,  but  also  in  those  cases  where  the  surface  is  very  nar- 
row or  even  discontinuous  ;  it  shows  us  how  diminution  in 
size  of  the  surface  produces  the  dilation  of  the  reflected  ray, 
and  how  a  system  of  very  narrow  mirrors  placed  side  by  side 
and  very  close  together  can  produce  colored  images,  owing  to 
the  mutual  influence  of  pencils  of  light  thus  dilated.  This  is 
the  phenomenon  of  ruled  surfaces.  With  the  same  ease  it  ex- 
plains the  images  and  colored  rings  produced  by  a  thin  fabric 
or  even  an  irregular  combination  of  very  fine  threads  or  small 
particles,  provided  they  are  almost  equal  in  size,  when  placed 
between  the  eye  of  the  observer  and  the  luminous  point. 

I  think  it  hardly  necessary  to  emphasize  these  phenomena, 
since  they  are  merely  combinations  of  those  described  above, 
and  since  I  have  attempted  to  give,  for  all  of  them,  a  general 
theory. 

144 


ON  THE  ACTION  OF  RAYS  OF  POLARIZED 
LIGHT  UPON  EACH  OTHER* 

BY 

ARAGO   ASTD   FRESNBL 


1.  Before  describing  the  experiments  which  form  the  sub- 
ject of  this  memoir  it  will  perhaps  be  well  to  recall  the  ex- 
quisite results  obtained  b^Dr.  Thomas  Young,  who,  with  rare 
sagacity  and  characteristic  skill,  has  already  studied  the  effects 
which  rays  of  light  exert  upon  each  other. 

First.  Two  rays  of  homogeneous  light  coming  from  the  same 
source  and  reaching  a  certain  point  in  space  by  paths  which 
are  different  and  slightly  unequal  in  length,  either  strengthen 
one  another  or  annul  one  another,  and  produce  upon  the  re- 
ceiving screen  a  bright  or  a  dark  point  according  as  the  differ- 
ence of  path  has  one  value  or  another. 

Second.  Two  rays  always  intensify  each  other  at  any  point 
tor  which  their  paths  are  equal ;  if  their  intensities  are  added 
foi  another  point  where  the  difference  of  path  is  equal  to  a 
quantity  d,  their  intensities  will  be  added  also  for  all  differ- 
ences of  path  included  in  the  series  2d,  3d,  etc.  The  interme- 
diate values  0-f  J^,  d+\d,  2d+%d,  etc.,  represent  the  points  in 
which  the  rays  annul  each  other. 

Third.  The  quantity  d  does  not  have  the  same  value  for  all 
homogeneous  rays.  In  air  its  value  for  the  extreme  red  rays  of 
the  spectrum  is  TTrffoo-  mm.,  while  for  violet  rays  it  is  only 
TFii_  mm.  For  other  colors  the  corresponding  values  are  in- 
termediate between  these  which  we  have  just  given. 

The  periodicity  of  color  which  is  seen  in  Newton's  rings,  in 

*  [Annale*  de  Chimie  et  de  Physique,  t.  x.,  p.  288  (1819).] 
K  145 


MEMOIRS    ON 

halos,  etc.,  seems  to  depend  upon  the  influence  exerted  upon 
one  another  by  rays  whose  paths  at  first  diverge,  and  later  are 
so  inclined  as  to  again  meet ;  but  in  order  to  bring  these  vari- 
ous phenomena  into  harmony  with  the  laws  just  stated,  we  are 
forced  to  adm.it  that  difference  of  path  alone  is  not  sufficient 
to  determine  the  mutual  action  of  two  rays  at  their  point  of 
meeting  except  when  they  are  both  travelling  in  the  same 
medium  ;  and  it  must  be  recognized  also  that  differences  of  re- 
fractive index,  or  thickness  in  the  transparent  bodies  traversed 
by  the  respective  rays,  produce  the  same  effect  as  difference 
of  path.  In  this  journal,  vol.  i.,  p.  199,  there  is  described  a 
direct  experiment  due  to  M.  Arago,  which  shows  the  same 
thing,  and  proves  also  that  a  transparent  body  diminishes  the 
speed  of  light  traversing  it  in  the  ratio  of  the  sine  of  the  angle 
of  incidence  to  the  sine  of  the  angle  of  refraction  ;  so  that  in 
all  the  phenomena  of  interference  *  two  different  media  produce 
similar  effects  when  their  thicknesses  are  in  inverse  ratio  to 
their  refractive  indices.  These  considerations  at  once  suggest 
a  new  method  for  measuring  slight  differences  of  refrangi- 
bility. 

2.  While  we  were  trying  to  determine  what  accuracy  was 
attainable  by  this  method,  one  of  us  (M.  Arago)  thought  that 
it  would  be  interesting  to  find  out  whether  the  actions  which 
ordinary  rays  exert  one  upon  another  were  in  any  way  modi- 
fied when  two  previously  polarized  pencils  of  light  were  made 
to  interfere.  We  know  that  if  a  narrow  body  be  illuminated 
by  light  coming  from  a  point-source,  its  shadow  is  bordered 
on  the  outside  by  a  series  of  fringes  produced  by  the  interfer- 
ence of  the  direct  light  with  the  rays  inflected  near  the  opaque 
body.  It  is  known  also  that  a  part  of  this  same  light,  passing 
into  the  geometrical  shadow  from  the  two  opposite  sides  of  the 
body,  there  gives  rise  to  fringes  of  the  same  kind. 

Now  the  fact  was  easily  recognized  that  these  two  systems 
of  fringes  are  absolutely  the  same,  whether  the  incident  light 
has  received  no  modification  whatever,  or  whether  it  has  been 
polarized  previous  to  incidence.  Rays  which  are  polarized,  in 
one  plane,  therefore  mutually  affect  one  another  in  the  same 
manner  as  rays  of  ordinary  light. 

*  Tliis  is  the  name  which  Mr.  Youug  has  given  to  the  phenomena  pro- 
duced by  the  meeting  of  two  or  more  rays  of  light. 

146 


THE    WAVE-THEORY    OF    LIGHT 

3.  It  was  still  to  be  determined  whether  two  rays  originally 
polarized  at  right  angles  would  not  produce  phenomena  of  the 
same  kind  when  they  met  inside  the  geometrical  shadow  of  an 
opaque  body.  For  this  purpose  we  placed  in  front  of  the 
point -source*  sometimes  a  rhomb  of  calc-spar,  sometimes  an 
achromatic  prism  of  rock  crystal,  and  thus  obtained  two  lumi- 
nous points.  In  each  case  we  had  a  divergent  pencil,  and'these 
two  pencils  were  polarized  at  right  angles.  Behind  the  two 
radiant  points  and  midway  of  the  space  between  them  was 
placed  a  cylinder  of  metal.  In  this  manner  a  part  of  the  po- 
larized light  from  the  first  pencil  reached  the  interior  of  the 
shadow  via  the  right-hand  side  of  the  cylinder;  while  a  part 
of  the  light  from  the  second  pencil,  polarized  in  a  plane  at  right 
angles  to  the  first,  entered  the  shadow  from  the  left-hand  side 
of  the  cylinder.  Some  of  these  rays  met  along  the  line  joining 
the  centre  of  the  cylinder  and  the  middle  point  of  the  straight 
line  drawn  from  one  luminous  point  to  the  other.  Here  these 
rays  had  traversed  equal  paths,  and  one  might  expect  them  to 
produce  fringes.  On  the  contrary,  not  the  slightest  trace  of 
fringes  could  be  seen,  even  with  a  magnifying-glass.  In  fact, 
the  rays  here  cross  without  either  affecting  the  other.  The 
only  fringes  which  make  their  appearance  in  this  experiment 
arise  from  the  interference  of  rays  which  come  from  only  one 
of  the  radiant  points  and  enter  the  shadow  from  each  side  of 
the  cylinder.  Those  which  we  tried  to  produce  by  the  inter- 
ference of  rays  polarized  at  right  angles  to  each  other  would  have 
occupied  a  position  intermediate  between  those  just  mentioned. 

Since  the  images  which  we  employed  were  not  very  widely 
separated,  the  thicknesses  of  crystal  traversed  by  the  ordinary 
and  the  extraordinary  rays  must  have  been  very  nearly  equal. 
Nevertheless,  similar  experiments  have  already  shown  us,  only 
too  frequently,  how  sensitive  the  phenomena  of  interference 
are  to  the  slightest  difference  of  speed  in  the  rays,  to  the 
length  of  path,  and  to  the  refractive  index  of  the  medium.  No 
argument  was  needed,  therefore,  to  convince  us  of  the  neces- 
sity of  repeating  these  experiments  under  conditions  which 
would  eliminate  these  various  sources  of  inaccuracy.  This  has 
been  attempted  by  each  of  us. 

*  For  all  the  experiments  described  in  this  paper  our  source  of  light  was 
the  focus  of  a  small  magnifying-glass. 

147 


MEMOIRS    ON 

4.  M.  Fresnel  at  once  devised  two  distinctly  different  meth- 
ods.    The  principle  of  interference  shows  us  that  pencils  of 
light  from  two  luminous  points,  originally  from  a  single  point- 
source,  produce  bright  and  dark  bands  at  points  of  intersection 
even  though  no  opaque  body  be  interposed.     (See  Annales  de 
Chimie  et  de  Physique,  t.  i.,  p.  332.) 

To  solve  the  problem  it  is  then  only  necessary  to  determine 
whether,  when  two  images  are  produced  by  placing  a  rhomb  of 
calc-spar  in  front  of  a  luminous  point,  they  will  behave  in  this 
same  way  ;  but  since,  from  the  theory  of  double  refraction, 
we  know  that  the  extraordinary  ray  traverses  carbonate  of 
lime  more  rapidly  than  the  ordinary  ray,  it  becomes  necessary 
to  compensate  this  extra  speed  before  the  two  rays  are  allowed 
to  intersect.  In  order  to  accomplish  this  a  method  was  em- 
ployed which  has  been  described  by  M.  Arago  in  this  journal, 
vol  i.,  p.  199.  M.  Fresnel  placed  in  the  path  of  the  extraor- 
dinary pencil  alone  a  plate  of  glass  whose  thickness  had  been 
determined  by  computation  in  such  a  way  that,  under  perpen- 
dicular incidence,  this  pencil  lost  nearly  all  the  ground  which, 
in  the  crystal,  it  had  gained  over  the  ordinary  ray.  By  slightly 
inclining  the  plate  the  compensation  could  be  made  exact.  In 
spite  of  these  precautions,  the  two  rays,  polarized  at  right 
angles,  gave  not  the  slightest  trace  of  interference  bands. 

In  another  experiment,  M.  Fresnel  compensated  for  the  dif- 
ference of  speed  in  the  two  rays  by  allowing  them  each  to  fall 
upon  a  small  unsilvered  mirror  whose  thickness  had  been  so 
computed  that  the  extraordinary  ray,  when  reflected  at  the  sec- 
ond face,  lost  by  twice  traversing  the  glass  more  than  it  had 
gained  in  traversing  the  crystal  ;  a  gradual  inclination  of  the 
plate  brought  about  complete  compensation. 

Under  no  angle  of  incidence,  however,  would  the  ordinary 
rays,  reflected  at  the  first  surface,  interfere  with  the  rays  re- 
flected from  the  second  surface  to  produce  bands. 

5.  In  order  to  avoid  the  theoretical  consideration  introduced 
into  the  preceding  experiment,  and  to  maintain  the  original 
intensity  of  the  light,  M.  Fresnel  adopted  the  following  meth- 
od :  A  rhombohedron  of  calc-spar  was  sawed  through  the  mid- 
dle, and  the  two  parts  were  placed  one  in  front  of  the  other 
with  their  principal  sections  at  right  angles.     In  this  position, 
the  ordinary  ray  from  the  first  crystal  was  refracted  as  an  ex- 
traordinary ray  in  the  second  ;  while,  conversely,  the  extraor- 

148 


THE    WAVE-THEORY    OF    LIGHT 

dinary  ray  in  the  first  crystal  suffered  ordinary  refraction  in 
the  second.  On  viewing  a  luminous  point  through  this  com- 
bination, one  sees  only  a  double  image.  Each  pencil  has  ex- 
perienced in  succession  the  two  kinds  of  refraction.  The  sum 
of  the  paths  of  each  pencil  through  the  two  crystals  ought, 
therefore,  to  be  equal,  since  by  hypothesis  the  crystals  have 
the  same  thickness  ;  so  that  everything  is  compensated,  both 
as  regards  speed  and  length  of  path.  Nevertheless,  two  systems 
of  rays  polarized  at  right  angles  never  gave  rise  to  any  interfer- 
ence fringes.  Lest  the  two  parts  of  the  rhombohedron  did  not 
have  quite  the  same  thickness,  we  took  pains  in  each  test  to 
vary  slightly  and  slowly  the  angle  of  incidence  at  the  face  of 
the  second  crystal. 

6.  The  method  devised  by  M.  Arago  for  solving  this  same 
problem  was  independent  of  double  refraction.  It  has  been 
known  for  a  long  time  that  if  one  cuts  two  very  narrow  slits 
close  together  in  a  thin  screen  arid  illuminates  them  by  a 
single  luminous  point,  there  will  be  produced  behind  the 
screen  a  series  of  bright  bands  resulting  from  the  meeting  of 
the  rays  passing  through  the  right-hand  slit  with  those  passing 
through  the  left.  In  order  to  polarize  at  right  angles  the  rays 
passing  through  these  two  apertures,  M.  Arago  at  first  thought 
of  using  a  thin  piece  of  agate,  sawed  through  the  middle  and 
placed  one  piece  in  front  of  each  slit,  in  such  a  way  that  the 
edges  formerly  meeting  along  the  line  of  the  cut  are  now  at 
right  angles  to  each  other.  This  arrangement  ought  certainly 
to  produce  the  effect  expected.  But  not  having  at  hand  a 
suitable  piece  of  agate,  M.  Arago  proposed  to  supply  its  place 
by  two  piles  of  plates,  of  proper  thickness,  built  up  from  sheets 
of  mica. 

For  this  purpose  we  selected  fifteen  plates  as  clear  as  possi- 
ble and  superposed  them.  This  pile  was  next  cut  in  two  by 
use  of  a  sharp  tool.  So  that  now  we  had  two  piles  of  plates  of 
almost  exactly  the  same  thickness,  at  least  in  those  parts  bor- 
dering on  the  line  of  bisection  ;  and  this  would  be  true  even  if 
the  component  plates  had  been  perceptibly  wedge-shaped. 
The  light  transmitted  by  these  plates  was  almost  completely 
polarized  when  the  angle  of  incidence  was  about  thirty  degrees. 
And  it  was  exactly 'at  this  angle  that  the  plates  were  inclined 
when  they  were  placed  in  front  of  the  slits  in  the  copper 
screen. 

149 


MEMOIRS    ON 

When  the  two  planes  of  incidence  were  parallel,  i.  e.,  when 
the  plates  were  inclined  in  -the  same  direction,  up  and  down, 
for  instance,  one  could  very  distinctly  see  the  interference 
bands  produced  by  the  two  polarized  pencils.  In  fact,  they  be- 
have exactly  as  two  rays  of  ordinary  light.  But  if  one  of  the 
piles  be  rotated  about  the  incident  ray  until  the  two  planes  of 
incidence  are  at  right  angles  to  each  other,  the  first  pile,  say, 
remaining  inclined  up  and  down  while  the  second  is  inclined 
from  right  to  left,  then  the  two  emergent  pencils  will  be  polar- 
ized at  right  angles  to  each  other  and  will  not,  on  meeting, 
produce  any  interference  bands. 

The  pains  we  took  to  make  these  two  piles  of  equal  thick- 
ness would  indicate  that  we  also  took  care  in  placing  them  be- 
fore the  slits  to  have  the  light  traverse  those  parts  which  were 
originally  contiguous.  But  all  difficulties  of  this  kind  are 
really  solved  by  the  fact  that  the  two  rays  when  polarized  in  the 
same  plane  interfere  like  ordinary  light.  Moreover,  we  could 
not  produce  interference  by  slowly  and  gradually  changing  the 
inclination  of  one  of  the  plates  so  long  as  the  planes  of  inci- 
dence were  at  right  angles. 

7.  The  same  day  that  we  tried  the  combination  of  these  two 
piles  we  also  tried  an  experiment  suggested  by  M.  Fresnel^  an 
experiment  which,  it  must  be  confessed,  is  less  direct  than  the 
preceding,  but  which  is  also  more  easily  performed  and  which 
demonstrates  equally  well  the  impossibility  of  producing  fringes 
by  bringing  together  rays  polarized  at  right  angles  to  each 
other. 

In  front  of  a  sheet  of  copper  in  which  are  cut  two  slits  we 
placed,  for  instance,  a  thin  plate  of  selenite.  Since  this  is  a 
doubly  refracting  crystal,  there  will  be  two  pencils  of  light 
polarized  at  right  angles  passing  through  each  slit.  Now  if 
rays  polarized  in  one  plane  can  affect  rays  polarized  in  a  plane 
at  right  angles,  we  should  expect  with  this  arrangement  to  see 
three  distinct  systems  of  fringes.  The  ordinary  rays  from  the 
right-hand  slit  would  combine  with  the  ordinary  rays  from  the 
left-hand  slit  to  form  a  first  system  symmetrical  with  respect 
to  the  line  bisecting  the  space  between  the  two  slits.  The 
bands  formed  by  the  two  extraordinary  pencils  would  fall  in 
the  same  position  as  the  preceding,  increasing  their  intensity, 
but  remaining  indistinguishable  from  them.  As  to  those  which 
would  result  from  the  action  of  the  ordinary  rays  from  the 

150 


THE    WAVE-THEORY    OF    LIGHT 

right  upon  the  extraordinary  from  the  left,  and  conversely,  it 
is  clear  that  they  would  form  a  system  to  the  right  and  to  the 
left  of  the  central  band.  The  distance  of  either  of  these  sys- 
tems from  the  centre  would  increase  with  the  thickness  of  the 
plate,  for,  as  we  have  seen,  difference  of  speed  is  quite  as  ef- 
fective as  difference  of  path  in  shifting  the  position  of  fringes. 
Now,  since  the  fringes  in  the  centre  are  the  only  ones  visible, 
even  though  the  plate  of  selenite  be  so  thin  as  not  to  shift  the 
other  two  systems  very  much,  we  must  conclude  that  rays  of 
light  polarized  at  right  angles  do  not  affect  one  another. 

8.  In  order  to  verify  this  conclusion,  suppose  that  we  cut 
the  selenite  plate  in  two,  and  that  we  place  one  half  in  front 
of  the  first  slit  and  the  other  half  in  front  of  the  other  slit; 
and  instead  of  placing  their  axes  parallel  as  in  the  case  of  a 
single  plate,  let  us  put  them  at  right  angles  to  each  other.     In 
this  way  the  ordinary  ray  coming  through  the  right-hand  slit 
will  be  polarized  in  the  same  plane  as  the  extraordinary  ray 
from  the  left-hand  slit,  and  vice  versa.     These  rays  will  then 
form  fringes  ;  but  their  speeds  in  the  crystal  will  not  be  equal, 
and   they  will  not  lie  symmetrically  about  the  middle  of  the 
space  between  the  two  slits.     Central  fringes  will  be  produced 
only  by  ordinary  or  extraordinary  rays  from  the  one  slit  meeting 
rays  from  the  other  slit  which  are  polarized  in  the  same  plane. 
But  when  the  two  parts  of  crystal  are  arranged  as  we  have 
here  supposed  them,  those  rays  which   are  polarized  at  right 
angles  to  each  other  ought  not  to  affect  one  another.     One 
would,  therefore,  see  simply  the  first  two  systems  of  fringes 
separated  by  an  interval  of  white  or  of  some  uniform  shade. 

[An  unimportant  foot-note  is  here  omitted.] 

If,  without  changing  the  experiment  in  any  other  respect, 
we  simply  set  the  two  plates  of  selenite  so  that  their  axes  make 
an  angle  of  45°  instead  of  90°,  we  should  at  once  see  three 
systems  of  fringes  ;  for  now,  since  their  planes  of  polarization 
are  no  longer  at  right  angles,  each  pencil  from  the  right  will  in- 
terfere with  the  two  pencils  from  the  left,  and  vice  versa.  It 
should  be  observed  also  that  the  middle  system  is  the  most  in- 
tense, resulting,  as  it  does,  from  the  exact  superposition  of  in- 
terference bands  of  polarized  pencils  of  the  same  kind. 

9.  Let  us  return  to  the  combination  of  the  two  piles  and 
imagine  that  the  planes  of  incidence  are  mutually  perpendicular, 
so  that  the  two  pencils  are  polarized  at  risjht  angles  to  each 

151 


MEMOIRS    ON 

other.  Between  the  copper  screen  and  the  eye  place  a  doubly 
refracting  crystal  in  such  a  way  that  its  principal  section 
makes  an  angle  of  45°  with  the  planes  of  incidence.  In  ac- 
cordance with  the  well-known  laws  of  double  refraction,  the 
rays  which  are  transmitted  by  the  piles  will  afterwards,  in  pass- 
ing through  the  crystal,  each  be  divided  into  two  others. 
These  two  will  be  of  equal  intensity,  will  be  polarized  in  planes 
which  are  mutually  perpendicular,  and  one  of  these  planes  will 
coincide  with  the  principal  section  of  the  crystal.  One  might 
therefore  expect  to  see,  in  this  experiment,  one  series  of 
fringes  due  to  the  meeting  of  the  ordinary  pencil  from  the 
right  with  the  ordinary  pencil  from  the  left,  and  a  second  se- 
ries similar  to  the  preceding,  but  arising  from  the  interference 
of  the  two  extraordinary  pencils.  Such,  however,  is  not  the 
case  ;  for  these  four  pencils  meet  and  produce  only  a  uniform 
illumination,  showing  not  the  slightest  interference.* 

This  experiment  shows  that  two  rays  originally  polarized  at 
right  angles  to  each  other  may  subsequently  be  brought  into 
the  same  plane  of  polarization  without  again  acquiring  the 
power  of  interference. 

10.  In  order  to  produce  interference  between  two  rays  po- 
larized at  right  angles  and  afterwards  reduced  to  the  same 
plane  it  is  necessary  that  they  should  originally  have  been  po- 
larized in  one  and  the  same  plane.  This  is  shown  by  the  fol- 
lowing experiment,  which  was  devised  by  M.  Fresnel. 

A  plate  of  selenite,  backed  with  a  sheet  of  copper  in  which 
two  apertures  have  been  made,  is  illuminated  by  a  pencil  of 
polarized  light  coming  from  a  point-source  and  striking  the 
selenite  plate  at  perpendicular  incidence.  The  axis  of  the 
plate  makes  an  angle  of  45°  with  the  original  plane  of  polariza- 
tion. As  in  all  similar  experiments  the  shadow  of  the  copper 
screen  is  observed  with  a  magnifying-glass  ;  but  in  this  case 

*  If  the  plate  interposed  between  the  copper  screen  and  the  eye  were  so 
thin  as  to  only  slightly  separate  the  two  images,  one  might  explain  the  ab- 
sence of  interference  as  follows  :  viz.,  suppose  the  two  systems  of  bands 
are  superposed  in  such  a  fashion  that  the  bright  bands  of  one  system 
coincide  with  the  dark  bands  of  the  other  system,  and  vice  versa.  But  the 
insufficiency  of  this  explanation  is  shown  by  placing  a  rhombohedron  of 
Iceland  spar  between  the  eye  and  the  preceding  crystal.  In  certain  posi- 
tions this  Iceland  spar  separates  the  two  systems  of  bands,  because  they 
are  polarized  at  right  angles.  But  even  under  these  circumstances  one 
sees  no  trace  of  bands. 

152 


THE    WAVE-THEORY    OF    LIGHT 

a  rhombohedron  of  Iceland  spar,  in  which  [the  separation  of 
images  due  to]  double  refraction  is  perceptible,  is  placed  in 
front  of  the  focus. 

The  principal  section  of  the  Iceland  spar  makes  an  angle  of 
45°  with  that  of  the  plate  of  selenite.  Accordingly  we  find  in 
each  image  three  systems  of  fringes,  one  falling  exactly  in  the 
middle  of  the  shaoow,  the  others  being  situated  on  the  right 
and  left  respectively. 

Let  us  now  consider  one  of  these  two  images,  say  the  or- 
dinary, and  see  what  gives  rise  to  these  three  systems  of 
bands. 

The  pencils  which  pass  through  the  two  slits  are  polarized 
in  the  same  plane,  but  on  emergence  from  the  plate  of  selenite 
they  are  divided  into  two  pencils  polarized  at  right  angles. 
Since  double  refraction  in  this  plate  is  inappreciable,  the  or- 
dinary and  extraordinary  pencils  each  follow  practically  the 
same  route,  though  with  different  speeds. 

Each  of  these  double  pencils,  say  the  one  from  the  right-hand 
slit,  will  be  divided,  in  passing  through  the  Iceland  spar,  into  four 
pencils,  two  ordinary  and  two  extraordinary ;  but,  as  a  matter  of 
fact,  one  will  see  only  two,  since  components  in  the  same  plane 
will  coincide.  It  is  also  evident,  from  the  well-known  laws  of 
double  refraction  and  from  the  relative  positions  of  the  selenite 
and  the  Iceland  spar,  that  at  emergence  from  this  latter  crys- 
tal the  ordinary  pencil  will  be  composed  partly  of  the  ray 
which  "was  ordinary  in  the  selenite  and  partty  of  the  ray  which 
was  extraordinary  ;  while  the  other  two  components  of  these 
same  rays  go  to  form  the  extraordinary  image  which  we  are 
not  now  considering.  The  pencil  which  emerges  from  the  left- 
hand  slit  behaves  in  the  same  way.  We  see,  in  fact,  that  the 
ordinary  pencil  coming  either  from  the  right  or  left  hand  slit 
will,  after  traversing  the  two  crystals  in  this  new  instrument, 
be  composed  partly  of  light  which  has  followed  the  ordinary 
path  in  each  crystal  and  partly  of  light  which  started  out  as  an 
extraordinary  ray. 

Eays  coming  from  the  two  slits  and  following  the  ordinary 
path  through  each  of  the  two  crystals  will  have  traversed 
routes  of  the  same  length  and  with  the  same  speed.  On  meet- 
ing, they  ought,  therefore,  to  give  rise  to  central  bands.  The 
same  is  true  of  rays  which  have  pursued  the  extraordinary 
path  both  in  the  selenite  and  in  the  Iceland  spar.  The  bands 

153 


MEMOIRS    ON 

in  the  middle  of  the  shadow  result,  therefore,  from  the  super- 
position of  these  two  different  systems. 

Now- as  to  that  portion  of  light  from  the  right-hand  slit 
which  has  traversed  the  selenite  as  an  extraordinary  ray,  for 
instance,  but  passed  the  Iceland  spar  as  an  ordinary  ray,  it  is 
evident  that  it  will  have  traversed  a  path  which  in  length  is 
equal  to  that  of  the  left-hand  pencil  which  made  the  whole 
trip  as  an  ordinary  ray.  But  since  in  the  selenite  the  speeds 
are  different,  those  points  where  they  meet  to  form  fringes  will 
not  lie  symmetrically  between  the  two  slits,  but  will  be  shifted 
to  the  right,  i.  e.,  to  the  side  opposite  the  ray  which  for  a 
while  travelled  as  an  extraordinary  ray,  but  now  travels  more 
slowly.  Finally,  as  a  last  combination,  we  have  interference 
between  that  component  of  the  right-hand  pencil  which  trav- 
ersed both  crystals  as  an  ordinary  ray  and  that  component  of 
the  left-hand  pencil  which  in  the  selenite  was  an  extraordinary 
ray  and  in  the  Iceland  spar  an  ordinary  ray.  This  interfer- 
ence gives  rise  to  a  system  of  bands  situated  on  the  left  of  the 
centre. 

We  have  now  explained  the  paths  of  the  rays  which  meet  to 
form  the  three  systems  of  fringes  in  the  experiment  under  discus- 
sion. And  it  may  be  remarked  that  the  right  and  left  systems 
were  produced  by  the  interference  of  rays  which  were  previous- 
ly polarized  at  right  angles  in  the  selenite  and  afterwards  re- 
duced to  the  same  plane  in  the  Iceland  spar.  Two  rays  polar- 
ized at  right  angles  and  later  reduced  to  the  same  plane-of  po- 
larization can,  then,  meet  and  produce  interference  bands ; 
but  for  this  purpose  it  is  an  essential  condition  that  the  rays 
should  ORIGINALLY  have  been  polarized  in  the  same  plane. 

So  far  we  have  not  considered  the  interaction  of  the  two 
pencils  which  suffered  extraordinary  refraction  in  the  Iceland 
spar.  These  pencils  also  furnish  three  systems  of  bands,  but 
they  are  separated  from  the  others.  If  we  allow  all  the  condi- 
tions of  the  experiment  to  remain  the  same,  except  that  we 
substitute  for  the  Iceland  spar  a  plate  of  selenite  or  quartz 
which  does  not  give  two  distinct  images,  the  six  systems,  in- 
stead of  being  reduced  to  three  by  superposition,  will  result  in 
one  central  system.  This  remarkable  fact  shows,  first,  that 
the  fringes  resulting  from  the  interference  of  the  ordinary 
rays  are  complementary  to  those  produced  by  the  interference 
of  the  extraordinary  rays  ;  and,  secondly,  that  these  two  sys- 

154 


THE    WAVE-THEORY    OF    LIGHT 

terns  are  so  located  that  a  bright  band  in  the  one  system  corre- 
sponds to  a  dark  band  in  the  other  system.  Were  these  two 
conditions  not  satisfied,  one  would  not  find  uniform  and  con- 
tinuous illumination  on  each  side  of  the  central  fringes.  We 
meet  here,  then,  the  same  difference  of  half  a  wave-length  that 
is  found  in  the  phenomena  of  colored  rings. 

From  the  experiments  just  described  we  may,  therefore,  infer 
the  following  facts  : 

(1.)  Two  rays  of  light  polarized  at  right  angles  do  not  pro- 
duce any  effect  upon  each  other  under  the  same  circumstances 
in  which  two  rays  of  ordinary  light  produce  destructive  inter- 
ference. 

(2.)  Rays  of  light  polarized  in  the  same  plane  interfere  like 
rays  of  ordinary  light ;  so  that  in  these  two  kinds  of  light  the 
phenomena  of  interference  are  absolutely  identical. 

(3.)  Two  rays  which  were  originally  polarized  at  right  an- 
gles may  be  brought  to  the  same  plane  of  polarization  -without 
thereby  acquiring  the  ability  to  interfere. 

(4.)  Two  rays  of  light  polarized  at  right  angles  and  after- 
wards brought  into  the  same  plane  of  polarization  interfere 
like  ordinary  light  provided  they  were  originally  polarized  in 
the  same  plane. 

(5.)  In  the  phenomena  of  interference  produced  by  rays 
which  have  experienced  double  refraction  the  position  of  the 
interference  bands  is  determined  not  only  by  difference  of  path 
and  difference  of  speed,  but  in  some  cases,  as  above  indicated, 
it  is  necessary  to  take  into  account  also  a  difference  of  one- 
half  a  wave-length. 

All  these  laws  are,  as  we  have  seen,  based  directly  upon  ex- 
perimental evidence.  In  starting  from  the  phenomena  of 
crystalline  plates,  they  can  be  derived  more  simply  ;  but  then 
we  have  to  assume  that  the  colors  of  the  plates  when  illumi- 
nated by  polarized  light  are  produced  by  the  interference  of 
several  systems  of  waves.  The  evidence  which  we  have  just 
presented  has  the  advantage  of  establishing  the  same  laws 
quite  independently  of  hypothesis. 

155 


MEMOIRS    ON 


BIOGRAPHICAL  SKETCH 

AUGUSTIN  JEAN  FRESNEL  was  born  in  Normandy  in  1788, 
and  died  near  Paris  in  1827. 

As  a  child  he  was  quite  the  reverse  of  precocious  ;  but  at 
the  age  of  sixteen  he  was  ready  to  enter  the  Ecole  Poly  tech- 
nique at  Paris,  where  he  received  sound  mathematical  training 
and  attracted  to  himself  the  attention  of  Legendre.  His  edu- 
cation was  completed  at  the  Ecole  des  Fonts  et  Ohausees  where 
he  received  an  engineer's  training.  Several  years  were  next 
spent  in  professional  work  in  various  parts  of  France. 

In  1816,  through  the  influence  of  Arago,  he  received  an  ap- 
pointment in  Paris,  where  he  remained  during  the  rest  of  his 
life.  When  we  recall  that  his  first  studies  in  optics  date  from 
1814,  his  accomplishments  during  the  eleven  years  of  his  Paris 
residence  must  ever  fill  us  with  wonder.  New  ideas  were  not 
only  rapidly  acquired,  but  were  also  rapidly  perfected.  They 
were  at  once  submitted  to  the  test  of  experiment  and  as  quick- 
ly received  elegant  mathematical  description. 

The  wave-theory  of  light  had  lacked  neither  merit  nor  able 
support ;  Grimaldi,  Hooke,  Efuygens,  and  Young  had  been  its 
advocates;  but  it  was  only  in  the  hands  of  Fresnel  that  the 
problem  and  its  solution  received  such  clear  and  simple  state- 
ment as  to  command  acceptance.  The  work  of  Fresnel  lies  ex- 
clusively in  the  domain  of  optics,  each  of  his  investigations  fall- 
ing into  one  of  two  distinct  groups,  viz.,  the  kinematics  of  light 
and  the  dynamics  of  light. 

His  earlier  papers  deal  with  questions  of  diffraction,  inter- 
ference, and  polarization,  in  which  the  chief  factors  of  the  dis- 
cussion are  displacements,  velocities,  and  squares  of  velocities 
— the  quantities  of  kinematics. 

His  later  papers,  however,  refer  more  to  the  medium  through 
which  luminous  energy  is  transferred;  they  deal  with  the  forces 
of  elasticity  here  brought  into  play,  and  seek  to  determine  the 
speed  of  light  as  a  function  of  the  mechanical  properties  of  the 
matter  through  which  the  light  travels ;  they  deal,  in  short, 
with  the  dynamics  of  light. 

But  the  particular  achievements  with  which  the  name  of 
Fresnel  must  always  be  associated  are 

156 


THE    WAVE-THEORY    OF    LIGHT 

(1.)  The  introduction  of  the  idea  of  transverse  vibrations. 

(2.)  The  combination  of  the  principle  of  Huygens  with  that 
of  interference. 

An  excellent  and  appreciative  sketch  of  Fresnel  will  be 
found  in  Arago's  Notices  BiograpMques,  vol.  i.  It  is  here  that 
he  paraphrases  Newton's  remark  concerning  Cotes  by  saying 
"  que  nous  savons  quelque  chose  quoique  Fresnel  ait  peu  vecu." 

Between  the  years  1866  and  1870  the  French  government 
published  the  works  of  Fresnel  in  three  worthy  quarto  vol- 
umes, ably  edited  by  Senarmont,  Verdet,  and  the  author's 
brother,  Leonor  Fresnel. 

157 


OOOOCfocTo     O^r-TT-T^i-^i-^— "'- 


BIBLIOGRAPHY 


HISTORICAL 

'Hooke.  Micrographia.     London,  1665. 

Posthumous  Works  of  R.  Hooke.     London,  1705. 
Priestley.  History  and  Present  State  of  Discoveries  concerning  Vision^ 

Light,  and  Colours.     2  vols.     London,  1772. 
JWilde.  Geschichte  der  Optik.     2  vols.    Berlin,  1838.     This  history 

covers  only  the  period  from  Aristotle  to  Euler. 

JVerdet  Lecons  d'Optique  Physique.     2  vols.     Paris,  1869.     The 

second  chapter — fifty  pages— is  devoted  entirely  to 
the  history  of  the  wave-theory. 
Introduction  aux  ceuvres  d'Augustin  Fi'esnel. 
See  CEuvres  (Completes  de  Fresnel,  t,  i.,  pp.  1-99. 

cLloyd.  Report  of  the  Progress  and  Present  State  of  Physical  Op- 

tics.    Brit.  Assoc.  Rep.  1834. 

cArago.  (Euvres  Completes.     Paris,  1854.      The  first  volume  con- 

tains valuable  biographies  of  Fresnel  and  Young. 
""Peacock.  Life  of  Thomas  Young.     London,  1855. 

„  Bosscha.  Christian  Huygens.     Rede  am  SOOstcn.     Gedachtnistage 

seines  Lebensendes.     Leipsig,  1895. 


DIFFRACTION  AND  INTERFERENCE 

--Grimaldi.  Physico-Mathesis  de  lumine,  coloribus,  et  iride.     Bononiae, 

1665.  Those  to  whom  Grimaldi's  work  is  not  acces- 
sible will  find  an  excellent  resume  of  his  observa- 
tions in  Priestley's  History. 

^Newton.  Opticks.     London,   1704.     Newton's  description   of   dif- 

fraction phenomena  (Bk.  III.)  and  of  the  behavior  of 
the  prism  (Bk.  II.)  should  be  read  by  every  student 
of  optics. 

Fraunhofer.          Neue  Modification  des  Lichtes,  etc.     1821 .     Translated  by 
Ames  in  Harper's  Scientific  Memoirs. 

Schwerd.  Beugungserscheinungen.     Mannheim,  1835. 

Stokes.  Dynamical  Theory  of  Diffraction.     Trans.   Camb.  Phil. 

8oc.,  9,  1  (1849).  R-printed  in  his  Math,  and  Phys. 
Papers,  vol.  ii.,  p.  243. 

Loramel.  Abh.  der  IT.  Cl.  der  Kon.  Bayer.  Akad.  der  Wiss.,  vol.  xv. 

160 


MEMOIRS    ON    THE   WAVE-THEORY    OF   LIGHT 

Cornu.  Journal  de  Physique,  3,  p.   1,  1874.     Interpretation  of 

Fresnel's  Integrals  in  terms  of  "  Cornu's  Spiral." 
Rayleigh.  Treatise  on  Sound,  Second  Edition,   vol.  i.     On  Group 

Velocity  of  Waves. 
Phil.  Mag.,  27,  460  (1889).     On  Interference  phenomena 

with  a  source  of  white  light. 

Schuster.  Phil.  Mag.,  31,  77  (1891).    Elementary  treatment  of  prob- 

lems in  diffraction. 
Phil.  Mag.,  37,  509  (1894).    Interference  phenomena  with 

a  source  of  white  light. 

Gouy.  Jour,  de  PJiysique,  p.  354  (1886).     On  Interference   phe- 

nomena with  a  source  of  white  light. 
Michelson.  Amer.  Jour.  Sci.,  39,  Feb.,  1890. 

Phil.  Mag.,  Mch..  1891. 
Phil  Mag.,  April,  1891. 
Phil.  Mag.,  Sept.,  1892. 
Comptes  rendus,  17th  April,  1893. 

Astronomy  and  Astrophysics,  12,  556(1893).  Compari- 
son of  meter  with  wave-length  of  cadmium  light. 
Light -waves  and  their  application  to  Metrology, 
Nature,  16,  Nov.,  1893. 

Rowland.  Gratings  in  Theory  and  Practice  ;  Astronomy  and  Astro- 

physics, 12,  129  (1893). 


ABERRATION 

vYoung.  Lectures  on  Natural  Philosophy,  vol.  i.,  p.  462. 

-Fresnel.  Ann.  de  Chimie  et  de  Physique,  9,  57  (1818). 

Stokes.  Phil.  Mag.,  27,  9  (1845)  ;  28,  76  (1846) ;  29,  6  (1846). 

Fizeau.  Ann.  de  Chimie,  [3]  57  (1859). 

Hoek.  Arch.  Neerlandaises,  3,  180  (1868). 

Airy.  Proc.  Roy.  Soc.,  2O,  35  (1872)  ;  21,  121  (1873).     Meas- 

urement of  the  aberration  constant  by  means  of  a 
telescope  whose  tube  is  filled  with  water. 

Michelson.  Amer.  Jour.  Sci.,  122,  120  (1881). 

Michelson  and  Morley.     Amer.  Jour.  Sci..,  131,  377  (1886). 
Phil.  Mag..  24,  449  (1887). 

Rayleigh.  Nature,  45,  499  (1892).    A  splendid  presentation  of  facts 

and  theories  up  to  1887. 

Glazebrook.  Report  on  Optical  Theories.     Brit.  Assoc.  Rep.  (1895). 

Lodge,  O.  Phil.  Trans.,  184,  727(1893). 

Pellat.  Jour,  de  Physique,  [3]    4,  21  (1895).     Discusses  case  of 

telescope  filled  with  water. 

Larmor  Ether  and  Matter.     Cambridge,  1900. 

STATIONARY  LIGHT- WAVES 

Zenker.  Lehrbuch  der  Photochromie.     Berlin,  1868. 

Rayleigh.  Phil.  Mag..  24,  158,  note  (1887). 

L  161 


MEMOIRS    ON    THE    WAVE-THEORY    OF   LIGHT 

Lippmanu.  Comptes  rendus,    112,  274  (1891);    114,  961    (1892); 

115,  575(1892). 
Nature,  46,  12. 

Becquerel.  Comptes  rendus,  112,  275  (1891). 

Wiener.  Wied.  Ann.,  4O,  203  (1890).    Shows  that  photographic 

effect  of  light-waves  is  due  to  vibration  of  electric, 
not  magnetic,  forces. 
Wied.  Ann.,  55,  225  (1895). 


SOME  IMPORTANT  TREATISES  ON  THE  WAVE-THEORY 

Knockenhauer.  Die  Undulationstheorie  des  Lichtes.     Berlin,  1839. 

Airy.  Undulatory  Theory  of  Optics.     London,  1866. 

Lord  Kelvin.  Molecular  Dynamics.     (Baltimore  Lectures.)    Baltimore 

1884. 

Ketteler.  Theoretische  Optik.     Braunschweig,  1885. 

Rayleigh.  Art.     Wave  Theory  of  Light.     Ency.  Brit  ,  1888. 

Mascart.  Traite  d'Optique.     Paris,  1889. 

Preston.  The  Theory  of  Light.     London,  1890. 

Kirchhoff.  Optik.     Leipzig,  1891. 

Basset.  A  Treatise  on  Physical  Optics.     Cambridge,  1892 

Poincare.  La  Lumiere.     2  vols.     Paris,  1892. 

Winklemann.  Handbuch  der  Physik.     Breslau,  1893. 

Ilelmholtz.  Electromagnetische  Theorie  des  Lichtes.     Hamburg,  1897. 


Gray  and  Matthews. 


Bessel's  Functions. 
fraction. 

162 


London,  1895.     Chapter  on  Dif- 


INDEX 


Aberration  (see  Preface). 
Arago,  148,  149,  157. 


Bosscha,  43. 


Cassini,  3. 

Color,  Newton's  Explanation  of,  52. 

Crested  Fringes  of  Grimaldi,  69. 


D 

De  la  Hire,  3. 

Descartes'  Idea  of  the  Ether,  11,  18, 
23. 

Diffraction  in  the  Shadow  of  a  Nar- 
row Obstacle,  63,  82. 

Diffraction  Past  an  Edge,  130. 

Diffraction  Through  a  Narrow  Aper 
ture,  114,  135. 

Diffraction  Through  Parallel  Slits, 
88. 

Diffraction,  Young's  Idea  of,  56 

E 

Effective  Ray,  113 

Emission  Theory  of  Newton,  Pres- 

nel's  Objections  to,  99. 
Ether,  Newton's  Idea  of,  49. 
Euler,  48. 


Fermat's  Principle,  40. 

Fresnel,  Biographical  Sketch  of,  156. 

Fresnel's  Integrals,  123. 

Fresnel's  Zones.  111. 

Fringes,  Interior  and  Exterior,  81. 


G 


Glazebrook,  161. 
Griiualdi,  69- 

H 

Herschel,  76. 
Hooke,  22  (see  Preface). 
Huygvns,  Biography  of,  42. 
Huygens's  Principle,  108,  118. 

I 

Integrals  of  Fresnel,  123. 

Intensity  of  Light,  120. 

Intensity  of  Vibration,  120. 

Interference,  146. 

Interference     of     Polarized    Light, 

145. 
Interference,    Young's    Explanation 

of,  60,  68 ;   Fresnel's  Explanation 

of,  101. 

L 

Larmor,  162. 

Light,  Intensity  of,  120. 

M 

Michelson.  162. 

Micrographia  of  Hooke  (see  Preface). 


Newton's  Optics,  48. 


Pardies.  22. 

Phase,  104,  143. 

Pioard.  15. 

Polarized  Light,  Interference  of,  145. 


163 


INDEX 


Rayleigh,  161. 
Reflection  of  Light,  25. 
Refraction  of  Light,  30. 
R5mer,  3,  13. 
Rowland,  161. 


Schuster,  162 

Secondary  Waves,  21,  143. 

Simple  Harmonic  Motion,  102. 

Speed  of  Light,  13. 

Stokes,  110. 


Trains  of  Waves,  20. 
Transparency,  31. 
Transverse  Vibrations,  156. 


Verdet,  121,  160. 
Vibration,  Intensity  of,  120. 

W 

Wave-Length  of  Light  as  Determined 
by  Fresuel,  128  ;  as  Determined  by 
Young,  71. 

\Vollaston.  76. 


Young's  Idea  of  Diffraction,  56 


i  2'ones  of  Fresnel,  Hi. 
164 


THE    END 


7  DAY  USE 

WHICH 

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